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Prizes versus Contracts as Incentives for
Innovation∗
Yeon-Koo Che,† Elisabetta Iossa,‡ Patrick Rey§
December 13, 2017
Abstract
Procuring an innovation involves motivating a research effort to generate a new
idea and then implementing that idea efficiently. If research efforts are unverifiable
and implementation costs are private information, a trade-off arises between the two
objectives. The optimal mechanism resolves the trade-off via two instruments: a mon-
etary prize and a contract to implement the project. The optimal mechanism favors
the innovator in contract allocation when the value of the innovation is above a certain
threshold and handicaps the innovator in contract allocation when the value of the
innovation is below that threshold. A monetary prize is employed as an additional
incentive but only when the value of innovation is sufficiently high.
JEL Classification: D44, H57, D82, O31, O38, O39.
Keywords: Contract rights, Inducement Prizes, Innovation, Procurement and
R&D.
∗We would like to thank Roberto Burguet, Estelle Cantillon, Pierre-Andre Chiappori, Daniel Danau,
Francesco Decarolis, Drew Fudenberg, Bruno Jullien, Laurent Lamy, Patrick Legros, Oliver Hart, Giancarlo
Spagnolo, Yossi Spiegel, Jean Tirole and the participants in several seminars and conferences for their useful
comments. We would also like to thank Vincenzo Mollisi for his assistance with data collection. We gratefully
acknowledge financial support from the European Research Council (ERC) under the European Community’s
Seventh Framework Programme (FP7/2007-2013) Grant Agreement N◦ 340903, and the University of Rome
Tor Vergata “Bando Doppia Cattedra” Program 2014-2016.†Department of Economics, Columbia University, [email protected].‡University of Rome Tor Vergata, IEFE-Bocconi and EIEF, [email protected].§Toulouse School of Economics, University of Toulouse Capitole, Toulouse, France,
1
1 Introduction
It is widely known that markets provide insufficient innovation incentives.1 One policy rem-
edy is to create demand for innovation via public procurement. Public buyers can use their
large purchasing power as a lever to spur innovation and boost the generation and diffusion
of new knowledge.2 Not surprisingly, governments devote substantial resources to procuring
innovative goods and services. In 2015, the US government spent approximately 21 billion
dollars on public R&D contracts and 19 billion dollars on defense R&D contracts. European
countries spent approximately 2.6 billion euros in 2011 for non-defense R&D procurement
alone (European Commission, EC 2014).3 Indeed, history is rich with examples in which
public procurement has had a major effect on the development and diffusion of innovations
such as supercomputers, large passenger jets, semi-conductors and the Internet.4
Procuring innovation presents a special challenge that is absent in the procurement of
standard products. The value of innovation is typically realized through a follow-on good or
service that embodies that innovation. Therefore, for procurement to be successful, not only
should there be a sufficient incentive for innovation ex ante but the selection of a supplier of
the follow-on project should also be efficient ex post. These two goals often conflict with each
other as the innovator need not be most adept at performing the follow-on project. Hence,
awarding the follow-on contract, say, to the most efficient supplier may not adequately
motivate the innovator. One solution is to shift the assignment of the follow-on contract so
as to favor or disfavor the innovator depending on the outcome of innovation. An alternative
is to award a cash prize to a successful innovator. Common wisdom suggests that a cash
prize would be a better instrument for incentivizing innovation as this mechanism does not
distort the assignment of the follow-on contract.
However, this simple wisdom is not borne out in practice as there is mixed the use of the
two instruments. Cash prizes are often offered in research contests and R&D procurements
but are less frequent for “unsolicited” innovation proposals, even though governments do
consider them and occasionally implement them. Procurement practices also vary in the
treatment of innovators in the follow-on projects. For unsolicited proposals, some countries
do not treat innovators differently at the follow-on contract stage, but other countries, such
as Chile, Korea, Italy, and Taiwan, give an advantage to the proposer/innovator at the
1Private incentives for innovation are insufficient because innovative activities often generate knowledge
that has significant positive externalities and is by nature difficult to protect via intellectual property.2Public procurement is a significant part of economic activity; for instance, it accounts for approximately
12% of GDP in OECD countries (OECD, 2017).3In parallel, the European Commission has adopted new directives (Directives 2014/24 and 2014/25)
that modernize the legislative framework on public procurement in order to incentivize public demand for
innovative goods and services.4See, e.g., Cabral et al. (2006) and Geroski (1990) for references.
2
contract award stage. Further, the degree to which the innovator is favored varies across
cases, presumably depending on the value of the proposed innovation. Even when a cash
prize is awarded to a successful innovator, this award does not preclude special treatment
of the innovator at the contract stage. For instance, R&D procurement is often bundled
with the procurement of a follow-on product, in which case, the winner of the R&D stage is
guaranteed to win the production contract. A case in point is EC’s “Innovation Partnerships”
model, under which research and production are procured through one single tender, with
the innovator also obtaining the contract for the production of the innovative project.5
The purpose of this paper is to study how the alternative instruments should be combined
to optimize the conflicting goals of innovation. We analyze this question by identifying an
optimal procurement mechanism in an environment where the procurer faces a moral hazard
problem ex ante and an adverse selection problem ex post. The risk-neutral innovator(s) first
undertake costly effort to innovate and then a supplier is selected to perform the project that
implements the chosen innovation. An innovator’s effort stochastically increases the value
of project, but this effort is unobserved by outsiders. Coupled with limited liability, this
gives rise to a non-trivial moral hazard problem. The value of innovation is realized before
a follow-on project is performed, and there are multiple suppliers, including the innovator,
who can perform this project. The cost of performing the project is private information,
which gives rise to an adverse selection problem. The value of innovation is verifiable, which
is realistic in many procurement settings, as we motivate later. This feature allows the
procurer to reward the innovators with prizes as well as the contract for implementing a
chosen project as a function of the observed values of innovations.
We first consider the case of a single innovator. Not only does this baseline model make
our insights transparent but it is also often relevant as many innovative projects procured by
public agencies are unsolicited and arrive one at a time. We first find that in the absence of
adverse selection, i.e., when the cost of performing the project is observed by the procurer,
the common wisdom is indeed valid: the optimal mechanism relies solely on a cash prize and
does not distort the assignment of the follow-on contract. Specifically, the buyer awards a
prize to the innovator whenever the value of innovation is above a threshold, and she assigns
the contract ex post to the supplier with the lowest cost of realizing that value.
The result is quite different, however, in the presence of adverse selection, i.e., when the
suppliers have private information about the cost of performing the follow-on project. The
private information generates rents for the supplier who performs the project. Rents that
accrue to the innovator can work as additional incentive or disincentive depending on the
5EC has an alternative model, “pre-commercial procurement” (PCP), whereby the public author-
ity procures R&D activities from the solution exploration to prototyping and testing phases but re-
serves the right to tender the newly developed products or services competitively. See EC (2007) and
https://ec.europa.eu/digital-agenda/en/pre-commercial-procurement.
3
value of innovation. Specifically, for a high enough value of innovation, the rents accruing
to the innovator constitute an incentive, thereby effectively reducing the shadow cost of
awarding the contract to the innovator relative to the other suppliers. Hence, the optimal
mechanism calls for biasing the assignment of the follow-on contract in favor of the innovator.
In contrast, for a low-value innovation, the rents accruing to the innovator constitute a
disincentive for innovation and thus raise the shadow cost of selecting the innovator for
the contract. In this case, the optimal mechanism calls for biasing the assignment of the
follow-on contract against the innovator.
Cash prizes can be part of the optimal mechanism but only as a supplementary tool.
Specifically, the optimal mechanism prescribes a cash prize to be awarded to the innovator
(only) when the value of innovation is so high that shifting the contract right is not sufficient
for fulfilling the incentive need. In a striking contrast to the common wisdom, therefore, a
contract right serves as a primary incentive tool, and a cash prize serves as a supplementary
tool (when the former does not meet the required incentive need).
Comparative statics reflect the same insights. When information rents are significant
(e.g., because costs are relatively heterogeneous or there are few potential suppliers), the op-
timal mechanism may rely solely on the contract right to incentivize innovation. In contrast,
when information rents are small (e.g., because costs are relatively homogeneous or there are
many potential contractors), or when the value of innovation is high compared with these
rents, the optimal mechanism involves a cash prize, again because a contract right alone is
not sufficient in that case.
We next extend the model to allow for multiple innovators. This situation is relevant for
R&D contests and procurements wherein the buyer has a clear sense about the desired type
of innovation and its feasibility. We show that the above insights carry over. First, contract
rights serve as a central tool for rewarding innovations. Specifically, the optimal mechanism
favors the proposer of a high-value project and disfavors the proposer of a low-value project
at the implementation stage. Second, as in the single innovator case, cash prizes serve as
a supplementary tool to be used only when an innovator’s project is particularly valuable
and/or when her research effort is particularly worth incentivizing. An interesting novel
finding is that when a cash prize is used, it is never split among multiple innovators. In this
sense, we establish a “winner-takes-all” principle for the allocation of a cash prize.
Our findings clarify several issues that are relevant to policy on public procurements.
First of all, as mentioned earlier, a longstanding question is whether contracting rights
should be allocated in such a way as to favor the innovator who originates the project.
Contrary to the received wisdom, we provide a clear rationale in the presence of moral
hazard and adverse selection for the use of contract right as an incentive for innovation.
Extreme examples of such situations are found in the bundling of the procurement of final
products with preceding R&D activities. The US Congress recently expanded the use of
4
Other Transaction Authority (“OTA”).6 OTA allows a prototype project to transition to a
follow-on production contract. The Department of Defense may make such award without
the use of competitive procedures provided that (i) competitive procedures were used in
the initial prototype transaction award and (ii) the OTA contractor successfully completed
the prototype project. Likewise, as mentioned earlier, the “innovation partnerships” model
introduced by EC allows for a similar bundling. In 2016, this model was announced by
Transport for London for the development of a new composite conductor rail system to fit
constrained areas of the underground network and to improve the energy efficiency.7
Second, our analysis provides some insight into the specific way in which a contract right
should be allocated. Our theory rationalizes the use of contract right as an incentive for
innovation and suggests that the degree to which the innovator is favored at the contract
awarding stage should depend both on the value of her innovation and on the degree of
uncertainty with regard to the cost of implementing the project. In practice, this balance
could be achieved by giving the innovator a bidding credit in the form of additional points in
the score of the firm’s bid in the tendering procedure. Such a system is adopted in Chile and
Korea to incentivize unsolicited proposals. Further, there is some evidence that the procurer
adjusts the bonus points based on project-specific characteristics. For example, in a Chilean
procurement of airport concessions, the first two unsolicited proposals obtained a bidding
bonus of 20 percent points of the allowed score, while the third airport proposal received 10
percent points.8 Other methods of favoring the innovator include the best and final offer
and Swiss challenge systems, to be discussed in detail later. We further note that for very
valuable projects, our optimal mechanism can be implemented by making the innovator a
prime contractor: the innovator receives a fixed price equal to the value of the project and
freely chooses whether to deliver the project itself or to subcontract it to a different firm.
Finally, our analysis also clarifies the role of monetary prizes for promoting innovation and
their optimal design. Specifically, we identify their roles as supplementary to the contract
rights. This role explains the relative paucity/insignificance of explicit monetary reward
given to unsolicited projects. Even when monetary prizes are used, for example, in research
contests, the prizes are usually not split across multiple winners. This feature is in turn
consistent with the winner-take-all principle that we establish.
The paper is organized as follows. In Section 2, we study the case of a single innovator.
Section 2.1 introduces the model, Section 2.2 presents a number of benchmarks, and Section
2.3 develops the main analysis. In Section 3, we extend the analysis to the case of multiple
6Section 815 of the National Defense Authorization Act for Fiscal Year 2016, available at:
https://www.congress.gov/bill/114th-congress/house-bill/1735.7London: Transport equipment and auxiliary products to transportation, periodic indicative notice –
utilities supplies, number 2016/S 217-395943.8These projects pertained to the expansion of the airports of Puerto Montt (June 1995), Iquique (August
1995) and Calama (October 1997); see Hodges and Dellacha (2007).
5
innovators. In Section 4, we discuss the insights that our analysis offers for the approaches
used in practice for unsolicited proposals and innovation procurement. In Section 5, we
discuss the related literature. In Section 6, we provide some concluding remarks.
2 Procurement with a Single Innovator
We consider here the case in which a single innovator may propose a project. This case serves
to explain the main results in a simple way, but it is also practically relevant for the case
with unsolicited proposals. The decision facing the buyer is whether to adopt the project
and, if so, how to select a contractor for its implementation.
2.1 Model
A buyer (e.g., a public agency) wishes to procure an innovative project in two stages: inno-
vation and implementation. In the first stage, an innovator, say firm 1, exerts effort e ≥ 0
to create a project. The effort e costs the innovator c(e) ≥ 0 but affects the value v of
the project stochastically. The innovation project is at least partly nonexcludable and non-
rivalrous, which makes the project non-commercializable. Hence, awarding an intellectual
property right is not an option.
We assume that c (·) is increasing, strictly convex, twice differentiable and such that
c′ (0) = 0. The value v is drawn from V := [v, v] according to a c.d.f. F (·|e), which admits
a twice-differentiable density f (·|e) in the interior. An increase in e shifts the distribution
F (·|e) in the sense of the monotone likelihood ratio property, that is:
f(v′|e′)f(v|e′)
>f(v′|e)f(v|e)
, for any v′ > v and e′ > e. (MLRP )
The innovator’s effort e is unobservable. The project value v is instead publicly observable
and verifiable. The verifiability of v is a reasonable assumption in many procurement contexts
in which projects can be described using precise functional and performance terms. For
example, in the case of technological improvements for faster medical tests, transport units
with lower energy consumption, or for information and communication technology (ICT)
systems with interoperability characteristics, v may capture speed of the medical test, the
degree of energy efficiency of the transport unit, or the technical functionalities of the ICT
system verified in submitted prototypes, respectively. Later, we explore the case in which
the project value v is not contractible and discuss how our insights can be transposed to
such situations (see Section 4.2).
In the second stage, n potential firms, including the innovator, compete to implement
the project. Each firm i ∈ N := {1, ..., n} faces a cost θi, which is privately observed and
6
drawn from Θ :=[θ, θ]
according to a c.d.f. Gi(·), which admits density gi(·) in the interior.
We assume that θ < v and that Gi(θi)/gi(θi) is nondecreasing in θi for each i ∈ N .
If the project is not implemented, all parties obtain zero payoff. If instead the principal
pays t to procure the project of value v, then her welfare is given by:
v − t.
By the revelation principle, we can formulate the problem facing the principal as that of
choosing a direct revelation mechanism that is incentive-compatible. A direct mechanism is
denoted by (x, t) : V × Θn → ∆n × Rn, which specifies the probability xi (v, θ) that firm i
implements the project and the transfer payment ti (v, θ) that it receives when the project
proposed by firm 1 has value v and firms report types θ := (θ1, ..., θn). By construction,
the assignment probabilities must lie in ∆n := {(x1, ..., xn) ∈ [0, 1]n |∑
i∈N xi ∈ [0, 1]}. The
dependence of the mechanism on the project value v reflects its verifiability, whereas the
absence of the argument e arises from its unobservability to the principal.
The timing of the game is as follows:
1. The principal offers a direct revelation mechanism specifying the allocation decision
(i.e., whether the project will be implemented and if so, by which firm) and a payment
to each firm as functions of firms’ reports on their costs.
2. The innovator chooses e, and the value v is realized and observed by all parties.
3. Firms observe their costs and decide whether to participate.
4. Participating firms report their costs; the project is implemented (or not), and transfers
are made according to the mechanism.
For each v ∈ V , let
ui(v, θ′i|θi) := Eθ−i [ti(v, (θ
′i, θ−i))− θixi(v, (θ′i, θ−i))]
denote the interim expected profit that firm i could obtain by reporting a cost θ′i when it
actually faces a cost θi, and let
Ui(v, θi) := ui(v, θi|θi)
denote firm i’s expected payoff under truthful revelation of its type θi. The revelation
principle ensures that there is no loss in restricting attention to a direct mechanism (x, t)
that satisfies incentive compatibility:
Ui(v, θi) ≥ ui(v, θ′i|θi), ∀i ∈ N,∀v ∈ V, ∀ (θi, θ
′i) ∈ Θ2. (IC)
7
Note that the principal cannot force the firms to participate before the project is de-
veloped by the innovator as they decide whether to participate only after learning their
cost. This is a natural assumption in many settings. For example, in the case of unsolicited
proposals, the identities of the candidates capable of executing the project are unknown
until the nature of the project – its value and the costs of implementing it – is determined.
This makes it difficult for the principal to approach prospective suppliers for early buy-in.
More generally, procurers are reluctant to enter firms into loss-making contracts as this ac-
tion would typically discourage the participation of risk-averse or small firms and also cause
severe service disruptions when a firm chooses to default rather than honor the contract.
Limited liability on the firm’s side is therefore reasonable in public procurement settings.
This feature requires the direct mechanism (x, t) to satisfy individual rationality :
Ui(v, θi) ≥ 0, ∀i ∈ N,∀v ∈ V, ∀θi ∈ Θ, (IR)
As we shall see, together with (IC), this requirement will cause the principal to leave infor-
mation rents to the selected supplier.9
We also assume that the principal must at least break even for each realized value v of
the project. In other words, a feasible mechanism (x, t) must satisfy limited liability :
Eθ [w (v, θ)] ≥ 0, ∀v ∈ V, (LL)
where
w (v, θ) :=∑i∈N
[xi (v, θ) v − ti (v, θ)]
denotes the principal’s surplus upon realizing the value v of the project. Limited liability
may arise from political constraints. Public projects are scrutinized by various stakeholders
such as legislative bodies, project evaluation authorities, consumer advocacy groups, and the
media, who might reject projects that are likely to incur a loss. We note however that it is
not crucial that the constraint is of the particular form assumed in (LL); the general thrust
of our analysis carries through as long as there is some cap on either the maximum loss that
the principal can sustain or the maximum payment that she can make to the firm.10 Indeed,
public agencies and local authorities tend to operate within the boundaries of well-defined
budgets. Procurement tenders typically include a contract value that specifies the maximum
payment allowed for the supplier.
9In the absence of this individual rationality constraint, the principal could leave no rents to the firms
by requiring them to “buy-in” to a contract via an upfront fee. As a result, the first-best outcome could be
achieved at the implementation stage, and there would be no gain from using contract rights to reward the
innovator; monetary prizes would indeed be preferable.10Without any such constraint, the optimal mechanism would not be well defined: the principal would
find it desirable to pay an arbitrarily large bonus to the innovator only for a vanishingly small set of projects
with values close to v. Such a scheme may be of theoretical interest but is unreasonable and unrealistic.
8
Finally, as the innovator chooses effort e in her best interest, the mechanism must also
satisfy the following moral hazard condition:
e ∈ arg maxe{Ev,θ [U1(v, θ1) | e]− c(e)} . (MH)
The principal’s problem is to choose an optimal mechanism satisfying these constraints.
More formally, she solves the problem:
[P ] maxx,t,e
Ev,θ [w (v, θ) | e] ,
subject to (IR), (IC), (LL) and (MH)
2.2 Benchmarks
Before solving [P ], it is useful to begin with two benchmarks.
No adverse selection ex post. In this benchmark, we shut off the adverse selection
problem by assuming that the principal observes the firms’ implementation costs. Formally,
the principal’s problem is the same as [P ], except that the constraint (IC) is now absent.
We label such a relaxed problem [P − FB], where “FB” refers to the first-best outcome. In
this problem, once the principal approves the project, she can implement it by paying the
true cost θi to firm i without giving up any information rent.
We show that incentivizing the research effort with contract rights is suboptimal in this
case. In line with conventional wisdom, cash prizes are then the best instrument as they
do not distort the selection of a supplier whereas contract rights do. Thus, the solution to
[P − FB] is characterized as follows:
Proposition 1. (First-Best) There exist λFB > 0 and eFB > 0 such that the optimal
mechanism solving [P − FB] awards firm i a contract to implement the project with the
following probability:
xFBi (v, θ) :=
{1 if θi < min {v,minj 6=i θj} ,
0 otherwise,
with a transfer that simply compensates the winning firm’s cost, except that firm 1 is paid
additionally a monetary prize equal to
ρFB(v) :=
{Eθ[∑
i∈N xFBi (v, θ) (v − θi)
]> 0 if v > vFB,
0 if v < vFB,
where vFB is such v < vFB < v and solves βFB(v) = 1, where
βFB(v) := λFBfe(v|eFB)
f(v|eFB),
9
and eFB satisfies (MH).
Proof. See Appendix A. �
The contract right is assigned efficiently to the firm with the lowest cost as long as it is
less than the value v of the project. Incentive for innovation is provided solely by the cash
prize in a manner familiar from the moral hazard literature (e.g., Mirrlees (1975); Holmstrom
(1979)). The realized project value v is an informative signal about the innovator’s effort,
and paying an additional dollar to the innovator for a project with value v relaxes (MH)
by fe(v|eFB)f(v|eFB)
. Multiplied by the shadow value λFB of relaxing (MH), βFB (v) = λFB fe(v|eFB)f(v|eFB)
measures the incentive benefit of paying an additional dollar to the innovator. The optimal
mechanism calls for awarding the maximal feasible prize to the innovator whenever the
project value v is high enough to indicate that the incentive benefit exceeds the cost (i.e.,
βFB (v) > 1) and awarding no prize otherwise. In the former case, (LL) must bind, so
the maximal feasible prize is given to the innovator from the net surplus that the project
generates after reimbursing the efficient firm for its cost of implementation.
In sum, the innovator’s incentive payment has a bang-bang structure: there exists a
threshold value such that the innovator obtains no prize when the value of project falls
short of that threshold, and a prize equal to the entire value of the project when it exceeds
that threshold.11 Importantly, absent adverse selection, the principal never uses contracting
rights to motivate the innovating firm.
No moral hazard ex ante. In this benchmark, we shut off the moral hazard problem
by assuming that the project value follows some exogenous distribution F (v) that does
not depend on effort. Formally, the problem facing the principal in this benchmark is the
same as [P ], except that the moral hazard constraint (MH) is absent and the distribution
function F (v|e) is replaced by the exogenous distribution F (v). The resulting problem,
labeled [P − SB], conforms to the standard optimal auction design problem, except for
the (LL) constraint. Ignoring the latter, the optimal auction is the standard “second-best
mechanism” familiar from Myerson (1981). One can easily see that this mechanism satisfies
(LL) and thus constitutes a solution to [P − SB] as well. As the associated analysis is
standard, we provide the characterization of the solution without a proof.
Proposition 2. (Myerson) The optimal mechanism solving [P−SB] is the standard second-
best mechanism, which awards firm i the contract to implement the project with probability:
xSBi (v, θi) :=
{1 if Ji(θi) ≤ min {v,minj 6=i Jj(θj)} ,
0 otherwise,
where Ji(θi) := θi + Gi(θi)gi(θi)
is firm i’s virtual cost.11This feature of the bang-bang solution is reminiscent of the results of several well-known works, such
as Mirrlees (1975) and Innes (1990).
10
2.3 Optimal Mechanism
We now consider problem [P ], in which the principal faces ex post adverse selection with
respect to firms’ implementation costs as well as ex ante moral hazard with respect to the
innovator’s effort. Throughout the analysis, we assume that an optimal mechanism that
induces an interior effort level e∗ exists. The following Proposition characterizes this optimal
mechanism:
Proposition 3. There exists λ∗ > 0 such that the optimal mechanism solving [P ] is char-
acterized as follows:
(i) The mechanism assigns a contract to firm i = 1, ..., n to implement the project with
probability
x∗i (v, θ) =
{1 if K∗i (v, θi) ≤ min
{v,minj 6=iK
∗j (v, θj)
},
0 otherwise,
where
K∗i (v, θi) :=
{Ji (θi)−min {β∗ (v) , 1} Gi(θi)
gi(θi)if i = 1
Ji (θi) if i 6= 1, and β∗ (v) := λ∗
fe(v|e∗)f(v|e∗)
.
(ii) The mechanism awards firm i an expected transfer, T ∗i (v, θi) = Eθ−i [t∗i (v, θ)], equal to:
T ∗i (v, θi) := θiX∗i (v, θi) +
∫ θ
θi
X∗i (v, s)ds+ ρ∗i (v),
where:
– the first term represents the expected cost of implementing the project by i,
– the second term corresponds to the information rent generated by the contract
assignment, and
– the third term corresponds to a “cash prize,” which is zero for a non-innovator
(i.e., ρ∗i (v) := 0 for i 6= 1) and, for the innovator (i = 1), is equal to
ρ∗1(v) :=
{Eθ[∑
i∈N x∗i (v, θ) [v − Ji(θi)]
]> 0 if β∗ (v) > 1,
0 if β∗ (v) < 1.
(iii) The effort e∗ satisfies e∗ > 0 and∫v
∫θ
[ρ∗1(v) +
G1(θ)
g1(θ)x∗1(v, θ)
]g(θ)dθfe(v|e∗)dv = c′(e∗).
11
Proof. See Appendix B. �
To gain more intuition about this characterization, it is useful to decompose the prin-
cipal’s payment to each firm (net of its cost) into two components. The first component
is the information rent that must be paid to elicit the firm’s private information about its
cost. By a standard envelope theorem argument, this component is uniquely tied to – and
should therefore be interpreted as being necessitated by – the awarding of the contract to
the firm. We can thus call this component the contract payment. The second component is
a constant amount paid to the firm regardless of its cost. As this component is not related
to the contract assignment, we call it the cash prize and denote it by ρ∗i (v). Obviously, the
principal would never pay any cash prize to a non-innovating firm (i.e., i 6= 1). For the
innovating firm (i.e., i = 1), however, a cash prize may be necessary. The question is how
the principal should combine these two types of payments to encourage innovation.
The key observation in answering this question hinges on the incentive benefit β∗(v) =
λ∗ fe(v|e∗)
f(v|e∗). As explained earlier, this term represents the value of paying a dollar to the
innovator for developing a project worth v – more precisely, the effect fe(v|e∗)f(v|e∗)
of relaxing
(MH) multiplied by its worth λ∗ to the principal. If moral hazard were not a concern, we
would have λ∗ = 0 and thus β∗ (v) = 0, and the optimal mechanism would reduce to the
second-best auction mechanism described in Proposition 2. But, in this second-best solution,
the innovator does not fully internalize the surplus that her effort generates for the buyer.12
Hence, (MH) binds at the optimum, implying that the optimal mechanism prescribes a
stronger incentive for effort than the second-best mechanism.
Given λ∗ > 0, the incentive benefit β∗(·) is nonzero, and the optimal mechanism departs
from the second-best mechanism. In particular, the contract assignment now depends on the
realized value of the project, through the shadow cost K∗i (v, θi). For a non-innovating firm
(i.e., i 6= 1), the shadow cost is the same as its virtual cost, Ji(θi), just as in the second-best
benchmark. Instead, for the innovator (i.e., i = 1), the shadow cost differs from its virtual
cost by a term, β∗(v)G1(θ1)g1(θ1)
, which reflects the need to incentivize its research effort.13
Intuitively, rewarding the innovator for a low-value project (evidence of low effort) weak-
ens the innovation incentives, whereas rewarding the innovator for a high-value project (evi-
dence of high effort) strengthens them. Indeed, by (MLRP ), β∗ (v) = λ∗ fe(v|e∗)
f(v|e∗)increases in
v, and there exists a unique v ∈ (v, v) such that β∗ (v) = 0. Thus, β∗ (v) < 0 when v < v, so
rewarding the innovator indeed weakens the innovation incentives. Awarding a cash prize to
the innovator is never optimal in this case. For the same reason, each dollar paid as informa-
12The innovator does have some incentives as her rents increase with effort. However, the resulting
incentives are not sufficient as these rents understate the surplus accruing to the buyer.13Awarding the contract to the innovator with type θ1 with an additional probability unit necessitates
giving information rent of a dollar to types below θ1—so G1(θ1)g1(θ1)
ex ante —and each dollar paid to the
innovator yields the incentive benefit of β∗(v).
12
tion rents weakens the innovator’s incentive, causing the shadow cost K∗1(v; θ1) of assigning
the contract to the innovator to exceed its virtual cost J1(θ1) by −β∗ (v)G1(θ1)/g1(θ1) (> 0).
Hence, compared with the second-best mechanism, when v < v, the optimal mechanism calls
for biasing the contract allocation against the innovator.
When instead v > v, there are two possibilities. If v < v := sup {v ∈ V | β∗ (v) ≤ 1},the incentive benefit β∗ (v) of paying a dollar to the innovator is positive but less than a
dollar. In this case, it is still optimal not to award a cash prize as this award would entail a
net loss for the principal. However, a fraction β∗ (v) of the information rent accruing to the
innovator goes toward its innovation incentive, which reduces the shadow cost K∗1(v, θ1) of
assigning the contract to the innovator below its virtual cost J1(θ1) by β∗ (v)G1(θ1)/g1(θ1).
Compared with the second-best benchmark, the optimal mechanism distorts the allocation
of the contract in favor of the innovator.
If v > v,14 a dollar payment to the innovator yields more than a dollar incentive benefit.
A cash prize is then beneficial, which is why ρ∗1(v) > 0. Hence, the principal transfers any
surplus she collects, either through the cash prize or through the information rent; that is,
(LL) is binding. Furthermore, any increase in information rents for the innovator simply
crowds out the cash prize by an equal amount. It follows that the incentive benefit of the
information rent paid to the innovator is at most one 1 (and not β∗ (v) > 1), and the
shadow cost K∗1(v, θ1) reduces to the actual production cost θ1. Compared with the second-
best mechanism, the optimal mechanism distorts the allocation of the contract in favor of
the innovator to the same extent that the principal would treat an “in-house” supplier. Any
further distortion in favor of the innovator is suboptimal because it decreases the total “pie,”
and thus the cash prize to the innovator, more than it increases the information rent to that
firm.
We can state these observations more formally as follows:
Corollary 1. There exist v and v with v < v < v ≤ v such that the optimal mechanism
has the following characteristics:
• If v < v, then no prize is awarded, and x∗1(v, θ) ≤ xSB1 (v, θ), whereas x∗i (v, θ) ≥xSBi (v, θ) for all i 6= 1;
• If v < v < v, then no prize is awarded, but x∗1(v, θ) ≥ xSB1 (v, θ), whereas x∗i (v, θ) ≤xSBi (v, θ) for all i 6= 1;
• If v > v (which only occurs if v < v), then a prize is awarded to the innovator and
x∗1(v, θ) ≥ xSB1 (v, θ), whereas x∗i (v, θ) ≤ xSBi (v, θ) for all i 6= 1.
14This case occurs only when β∗ (v) > 1 so that v < v.
13
Whether it is optimal to award a monetary prize (i.e., v < v) depends on how much effort
needs to be elicited from the innovator and on how much incentive could be provided by
the information rents under a standard second-best auction. We show in Online Appendix A
that awarding a prize can be optimal when the support of project values is large and when
there is either little cost heterogeneity or a large number of firms. The former indicates that
the moral hazard problem is significant, whereas the latter implies that the procurement
auction does not generate much in information rents.
Corollary 1 shows that the optimal mechanism departs from a standard second-best
auction in different ways for high-value and low-value projects. In fact, this mechanism can
be easily implemented as a variant of common procurement designs.
• v > v: Bidding credit. In this range, the contract allocation is biased in favor of
the innovator, who may be selected to implement the project even when it is not the
most efficient firm. In practice, this result could be achieved by giving the innovator a
bidding credit in the tendering procedure. Bidding credits can take many forms, but
most commonly, they consist of additional points in the score of the firm’s bid. Such
a system is adopted in Chile and Korea to incentivize unsolicited proposals.
• v < v: Handicap. In this range, the contract allocation is biased against the innovator,
who may not be selected to implement the project despite being the most efficient firm.
We are not aware of the use of such a bias for procuring innovative projects; however,
handicap systems are used, for example, when governments want to favor domestic
industries.15 We discuss this further below (see section 4.1).
We note further that in the region where a cash prize is optimal, the mechanism can be
implemented in a very familiar and simple manner:
• v > v: Full delegation. In this range, the innovator is awarded a monetary prize
ρ∗1 (v) equal to the full value of the project (net of the cost and the information rents)
and obtains the contract if θ1 < min {v,mini 6=1 Ji(θi)}. This result can be achieved
by delegating the procurement to the innovator for a fixed price equal to the value
of the project. Indeed, suppose that the principal offers a payment v to the innova-
tor to deliver the project either by itself or by subcontracting it to a different firm.
The innovator then acts as a prime contractor with the authority to assign produc-
tion. Under this regime, facing the price v > v and given θ1, the innovator chooses
15Under “preferential price margins”, purchasing entities accept bids from domestic suppliers over foreign
suppliers as long as the difference in price does not exceed a specific margin of preference. The price preference
margin can result from an explicit “buy local policy,” e.g., the “Buy America Act.”
14
(x(v, ·), t(v, ·)) : Θn → ∆× Rn−1 so as to solve:
maxx,t
Eθ−1
[(v − θ1)x1(v, θ1, θ−1) +
∑i 6=1 {vxi(v, θ1, θ−1)− ti(v, θ1, θ−1)}
],
subject to (IR) and (IC)
The standard procedure of using the envelope theorem and changing the order of
integration results in the optimal allocation x solving
maxx,t
Eθ−1
[(v − θ1)x1(v, θ1, θ−1) +
∑i 6=1
[v − Ji(θi)]xi(v, θ1, θ−1)
],
which is exactly the allocation x∗ for the case of v > v.
The above results also have implications for project adoption itself. For instance, when
only the innovator can implement the project (n = 1), our results simplify to:
Corollary 2. For n = 1, we have:
• If v < v, then K(v, θ) > J(θ) (> θ): Compared with the first-best mechanism, there
is a downward distortion – under-implementation of the project – that is even more
severe than that in the standard second-best mechanism.
• If v < v < v, then J(θ) < K(v, θ) < θ: There is still a downward distortion compared
with the first-best mechanism, but it is less severe than in the standard second-best
mechanism.
• If v ≥ v, then J(θ) < K(v, θ) = θ: There is no distortion anymore; the project is
implemented whenever it should be from a first-best standpoint.
Illustration. To illustrate the above insights, consider the following example: (i) imple-
mentation costs are uniformly distributed over Θ = [0, 1]; (ii) the innovator can exert an
effort e ∈ [0, 1] at cost c (e) = γe; and (iii) the value v is distributed on [0, 1] according to
the density f (v|e) = e+ (1− e) 2 (1− v): exerting effort increases value in the MLRP sense
from a triangular density peaked at v = 0 for e = 0 (in particular, f (1|0) = 0) to a better (in
fact, the uniform) distribution for e = 1. Note that fe (v|e) = 2v − 1 ≷ 0⇐⇒ v ≷ v = 1/2.
The linearity of the cost and benefits ensures that it is optimal to induce maximal effort
(e∗ = 1) as long as the unit cost γ is not too high. Conversely, as long as e∗ = 1, the
Lagrangian multiplier λ∗ increases with the cost γ. For exposition purposes, we will use
different values of λ∗ (reflecting different values of γ) to illustrate the role of innovation
incentives.
15
v v
θ
(a) “small” λ
v v v
θ
(b) “large” λ
Figure 1 – Project implementation under different values of λ.
Consider first the case in which only the innovator can implement its project (i.e., n = 1).
Figure 1 depicts the range of the firm’s costs for which the project is implemented under
the optimal contract for different project values v. Figure 1-(a) depicts the case of λ∗ = 0.8,
where v = v, implying that a monetary prize is never awarded, whereas Figure 1-(b) shows
the case of λ∗ = 4, where 0 = v < v < v = 5/8 < v = 1. As the cost is uniformly
distributed, the highest cost for which the project is implemented also equals the probability
of the project being implemented, p∗ (v) := Eθ [x∗1(v, θ)]. Compared with the second-best
mechanism, depicted by the dashed line, the optimal mechanism implements the project for
a smaller range of costs (thus with a lower probability) when the project has a low value
(v < v = 1/2) but implements the project for a larger range of costs (and thus, with a
higher probability) when the project has a high value (v > v). When λ∗ = 4 (Figure 1-(b)),
stronger innovation incentives are required for large project values: there is a range of values
v > v for which (LL) is binding such that the principal exhausts the use of contract rights
as an incentive for innovation and begins offering a cash prize. As noted, in such a case, the
optimal assignment coincides with the first-best mechanism, depicted by the 45-degree line.
Focusing on the case λ∗ = 4, Figure 2 illustrates the case of n = 2 when the cost of each
firm is distributed uniformly over Θ = [0, 1]):
• For v < v (see Figure 2-(a), where v = 1/4): Compared with the first-best or the
standard second-best mechanisms, it is again optimal to bias the allocation of the
contract against the innovator. This result is now achieved in two ways. As before, the
project is implemented less often than in the second-best (and thus, a fortiori, in the
first-best) mechanism: the optimal mechanism shifts the vertical boundary of project
16
Firm 1
Firm 2
116
18 θ1
18
14
θ2
(a) v < v
Firm 1
Firm 2
724
715 θ1
724
712
θ2
(b) v < v < v
Firm 1
Firm 2
45
25 θ1
45
25
θ2
(c) v > v
Figure 2 – Contract assignment under different values of λ.
implementation to the left of the second-best mechanism (depicted by the dashed
line). In addition, however, when the project is implemented, the innovator obtains
the contract less often than in the first-best or the standard second-best mechanisms,
where the more efficient supplier would be selected; graphically, this bias is represented
by the triangular shaded area.
• For v < v < v (see Figure 2-(b), where v = 7/12): Compared with the standard
second-best mechanism (depicted by the dashed lines), it is now optimal to reward the
innovator both by implementing the project more often (rectangular shaded area) and
by favoring the innovator in the competition with their rival (triangular shaded area).
• Finally, for v ≥ v (see Figure 2-(c), where v = 4/5), the innovator’s shadow cost reduces
to its actual cost. The allocation of the contract thus favors the innovator even more,
and the project is implemented substantially more often than in the standard second-
best mechanism (specifically, it is now implemented whenever θ1 < v), although it
is implemented less often than in the first-best mechanism (e.g., when θ2 < v < θ1
and J (θ2) = 2θ2). Graphically, the rectangular and triangular shaded areas further
expand.
3 Procurement with Multiple Innovators
We now assume that several firms may innovate and propose projects as well as implement
them. This case captures the problem of a buyer who wishes to procure innovative projects,
17
products or services that several firms are capable of developing. The buyer has a clear sense
of what she needs, but an innovation is necessary to fulfill her demand. Examples include
the Norwegian Department of Energy procuring a new technology for carbon capture and
storage;16 or the Scottish Government procuring low-cost, safe and effective methods of
locating, securing and protecting electrical array cables in Scottish sea conditions.17 In both
instances, the public authority called for projects by means of requests for proposals (RFP)
with detailed specifications, and multiple firms responded by submitting different projects.
For the sake of exposition, we will suppose from now on that each firm k = 1, ..., n
can develop a project of value vk, which is publicly observable and distributed over V ac-
cording to a c.d.f. F k(vk|ek) with density fk(vk|ek), where ek denotes firm k’s innovation
effort.18 As before, effort ek costs firm k ck(ek), where ck(·) is increasing, strictly convex,
twice differentiable and such that ck′ (0) = 0. We assume that firms decide on these efforts
simultaneously, and we denote by e = (e1, ..., en) the profile of efforts chosen by them. The
alternative projects correspond to competing ways of fulfilling the same need, so they are
substitutes in the sense that the buyer will choose at most one project. The previous setting
corresponds to the special case where F k is concentrated on v for all k 6= 1.
In practice, a firm’s cost of implementing a project may depend on the nature of inno-
vation, including the identity of the innovator. In some cases, the innovator may have cost
advantages in implementing the project, for example, because of its superior knowledge of
the proposed solution. In other cases, the innovator may have cost disadvantages, for ex-
ample, because it is specialized in R&D and lacks the manufacturing capabilities necessary
to implement the developed prototype. To accommodate such an interdependency between
innovation and implementation, we assume that firm i’s cost of implementing project k is
given by θi + ψki , where:
• as before, θi is an idiosyncratic shock that is privately observed by firm i and distributed
according to the c.d.f. Gi;
• ψki represents an additional cost, potentially both project- and firm-specific, which for
simplicity is supposed to be common knowledge.
Without loss of generality, we consider a direct revelation mechanism that specifies an
allocation and a payment to each firm as a function of realized project values, v = (v1, ..., vn),
16Contract notice n. 2011/S 129-214787, ”NO-Porsgrunn: services related to the oil and gas industry”,
Supplement of the Official Journal of the European Union.17Contract notice n. 2013/S 249-436615, ”United Kingdom-Glasgow: Marine research services”, Supple-
ment of the Official Journal of the European Union.18While formally, all implementors are also innovators, the case of “pure contractors” can be accommo-
dated by setting the density to zero for v > v.
18
and of reported costs. Note that an allocation involves a decision as to which project is
selected as well as who implements that project.
A mechanism is thus of the form (x, t) : V n × Θn → ∆n2 × Rn. The objective of the
principal can now be expressed as:
maxx,t,e
Ev,θ [w (v, θ) | e] ,
where the ex post net surplus is now equal to
w (v, θ) =∑i∈N
[∑k∈N
vkxki (v, θ)− ti (v, θ)
].
The individual rationality and incentive compatibility constraints become
Ui(v, θi) ≥ 0, ∀i ∈ N,∀v ∈ V n,∀θi ∈ Θ,
Ui(v, θi) ≥ ui(v, θ′i|θi), ∀i ∈ N,∀v ∈ V n,∀ (θi, θ
′i) ∈ Θ2,
where firm i’s interim expected profits when lying and when reporting the truth are respec-
tively given by:
ui(v, θ′i|θi) = Eθ−i [ti(v, θ′i, θ−i)−
∑k∈N
(θi + ψki )xki (v, θ′i, θ−i)] and Ui(v, θi) = ui(v, θi|θi).
Finally, the buyer’s limited liability and firms’ moral hazard constraints can be expressed as:
Eθ [w (v, θ) | e] ≥ 0, ∀v ∈ V n,
ei ∈ arg maxei
Ev,θ
[Ui(v, θi) | ei, e−i
]− ci(ei), ∀i ∈ N.
As in the previous section, we assume that an optimal mechanism exists that induces
an interior profile of efforts ek(given our assumptions on the cost functions). The following
Proposition then partially characterizes this optimal mechanism:
Proposition 4. There exists λ∗ = (λ1∗, ..., λn∗) ≥ 0 such that the optimal mechanism solving
[P ]:
• selects firm i to implement project k with probability
xk∗i (v, θ) =
{1 if vk −K∗i (v, θi)− ψki ≥ max
{0,max(l,j)6=(k,i) v
l −K∗j (v, θj)− ψlj}
,
0 otherwise,
where
K∗i (v, θi) := Ji(θi)−(
βi(vi)
max{maxk βk(vk), 1}
)(Gi(θi)
gi(θi)
), and βi∗(vi) := λi∗
f ie(vi|ei∗)
f i(vi|ei∗).
19
• awards each firm i an expected transfer
T ∗i (v, θi) := ρ∗i (v) +∑k∈N
(ψki + θi
)Xk∗i (v, θi) +
∫ θ
θi
∑k∈N
Xk∗i (v, s)ds,
where Xk∗i (v, θi) = Eθ−i [xk∗i (v, θi, θ−i)]; and the transfer includes a cash prize
ρ∗i (v) := Eθ
[∑k,j∈N
xk∗j (v, θ){vk − ψkj − Jj(θj)
}],
which is positive only if βi∗(vi) > max {maxj∈N βj(vj), 1}.
Proof. See Online Appendix B. �
To interpret this characterization, consider first the case in which known differences in im-
plementation cost are additively separable across suppliers and projects: ψki = ψi+ψk for all i
and k. Then, the project selection is simply based on the “net values” of the projects, vk−ψkwithout regard for who implements the chosen project.19 While the selection of project is
largely independent of the identity of the supplier who would implement the chosen project,
the selection of project does depend on the realized values of the project. Specifically, the
K∗i (v, θi) depend on all realized values, including those of unselected projects.20 In particu-
lar, a higher vk calls for increasing not only the probability that project k is selected, but
also the probability that firm k obtains the contract to implement the chosen project even
when project k is not selected.
If the separability condition is not satisfied, the choices of the project and of the supplier
are more closely linked. For instance, suppose that ψkk = 0 < ψki = ψ for all k and i 6= k;
that is, each firm has a cost advantage of ψ for the project it proposes. If two firms i and
j are such that vi > vj, but θj is significantly lower than θi, the desire to exploit this cost
advantage may lead the principal to choose project j over project i.
A few other observations are worth making. First, as intuition suggests, the optimal
allocation xk∗i (v, θ) is nondecreasing in (vi, θ−i) and nonincreasing in (v−i, θi). In addition,
as all firms are now potential innovators, each virtual cost K∗i (v, θi) is characterized by two
cutoffs, vi and vi, which are defined as in the previous section but with somewhat different
implications. As before, each innovator is favored by a bias at the implementation stage
when vi > vi := β−1i (0) and is instead handicapped when vi < vi. To what extent a firm will
19To see this fact, note that the difference in surplus when a contractor i implements project k or project
l is given by (vk −Ki − ψk − ψi
)−(vl −Ki − ψl − ψi
)=(vk − ψk
)−(vl − ψl
),
and thus does not depend on which contractor i is selected.20Note that for a “pure contractor,” Ki (v, θi) = Ji (θi), as in a standard second-best auction.
20
be favored or handicapped depends on the relative magnitude of the shadow values βi(vi)
across firms and thus on the values of the other projects, v−i.
Second, a “winner-takes-all” principle holds in the sense that at most one firm is awarded
a cash prize. As in the case of a single innovator, a cash prize is worth giving only when
the incentive benefit βi (vi) exceeds one. However, with multiple innovators, there may be
several firms i for which βi (vi) > 1. Due to the limited liability of the buyer, an additional
dollar paid to a firm is one less dollar available to reward another firm. As the incentive
benefit of a dollar is proportional to βi (vi), the marginal benefit of the prize is maximized
by giving the prize only to the firm with the highest βi (vi). Splitting the available cash
across firms is never optimal for the same reason that it was never optimal to give less than
the maximal prize to the innovator in the single innovator case.
Third, in case a prize is used, it should be given to the firm the effort of which was worth
incentivizing most, that is, to firm ı := argmaxi∈N {βi (vi)}. Furthermore, only that firm
will face undistorted virtual cost K∗i (v, θi) = θi; the others will face a distorted virtual cost
equal to
K∗i (v, θi) = θi +
[1− βi (vi)
β ı (v ı)
]Gi (θi)
gi (θi)> θi.
Note that the recipient of the prize need not be the firm with the best project (i.e., the
highest vi). In the same vein, the recipient of the prize is not necessarily the firm to which the
selected project belongs. For instance, if innovators have the advantage of implementing their
own projects, but firms are otherwise ex ante symmetric (so that βi (·) = β (·)), then the firm
with the best project may receive a prize (if the value of its projects exceeds v = β−1 (1)),
and yet cost considerations may lead the principal to select another project.21
Finally, Proposition 4 does not explicitly characterize the set of firms that are induced to
innovate. Since losing projects are never implemented, innovation efforts are “duplicated.” If
there is little uncertainty in the outcome of innovation (e.g., the fi’s are highly concentrated
on narrow supports), then it may be optimal to induce only one firm to exert efforts. This is
entirely consistent with Proposition 4: λi∗
will be strictly positive for only one firm i in that
case. An asymmetric treatment of firms may arise even when firms are ex ante symmetric
(i.e., ψki = ψ and fk (·) = f (·)), since λ∗ = (λ1∗, ..., λn∗), and thus, β = (β1, ..., βn) could be
asymmetric endogenously. In this case, the mechanism would call for treating even ex ante
symmetric firms asymmetrically.
In practice, however, there are a couple of reasons why multiple firms may be induced to
innovate. First, if the outcomes of innovation efforts are sufficiently uncertain and stochastic,
then there is a “sampling” benefit from inducing multiple firms to make efforts and generate
21Consider for example the case n = 2, and suppose that ψ11 = ψ2
2 = 0 < ψ12 = ψ2
1 = +∞ (that is, a firm
can only implement its own project). In this case, if v1 > max{v2, v
}but θ1 < θ2, firm 1 receives a prize,
but firm 2’s project is selected if the cost difference is large enough.
21
favorable draws. Second, a procurer, particularly in the public sector, is often prohibited
from discriminating against firms, especially when they are ex ante symmetric. Proposition
4 is valid even for this case, and we will have λ1∗ = ... = λn∗. In that case, the best project
is selected, and from MLRP, the highest βi (vi) corresponds to the highest vi; hence, only
that project can ever be awarded a cash prize.
Remark 1 (Self-serving innovation strategies). We have so far assumed that firms’ R&D
efforts affect only the values of their projects. In practice, an innovator may choose from
a range of projects that differ in terms of the relative cost advantage the innovator would
enjoy vis-a-vis the other firms. In that case, the innovator may have an incentive to target
an innovation project that they will be best positioned to implement. For instance, a firm
may entrench itself by pursuing an innovation project that no other firm can implement.
Such a targeting possibility would further reinforce the main thrust of our results. While the
second-best auction would actually encourage such self-serving innovation strategies, favoring
innovators with high-value projects, as prescribed by our optimal mechanism, would mitigate
these incentives and encourage instead the adoption of more valuable innovation strategies.]
4 Discussion
In this section, we discuss how our insights relate to the mechanisms used in practice. We first
consider some feasibility issues with respect to handicaps (Section 4.1) and the verifiability
of the value of proposals (Section 4.2). We then discuss the implications of our analysis
for current practice in the management of unsolicited proposals (Section 4.3) and in the
procurement of innovation (Section 4.4).
4.1 On the feasibility of handicaps
The optimal mechanism relies on a “stick and carrot” approach, rewarding good proposals
by conferring an advantage in the procurement auction (possibly together with a monetary
prize) and punishing weak proposals with a handicap in the procurement auction. In practice,
many innovation procurement mechanisms employ cash prizes and/or favorable contract
assignments as incentives, but handicaps for weak projects do not appear to be used. The
lack of handicaps may stem from the risk of manipulation: an innovator with a low-value
project may, for instance, get around the handicapping by setting up a separate corporate
entity to participate in the implementation tender.
To gain some sense of how the mechanism would need to be adjusted if handicaps were
explicitly ruled out, in Online Appendix C, we extend our baseline model by assuming that
22
the innovator cannot be left worse off than under the standard second-best mechanism.22
That is, the mechanism must take into account the additional constraint:
x1(v, θ) ≥ xSB1 (v, θ).
We show that when keeping constant the multiplier λ for the innovator’s incentive constant,
ruling out handicaps has no impact on the contract right for high-value projects (namely,
those with v ≥ v), as x∗1(v, θ) > xSB1 (v, θ) in this case. In contrast, for low-value projects (i.e.,
those with v < v), the no-handicap constraint is binding, and the innovator’s probability of
obtaining the contract is increased from x∗1(v, θ) to xSB1 (v, θ). Interestingly, the no-handicap
constraint does not affect the size of the prize. Of course, removing the “stick” raises the cost
of providing innovation incentives, and thus, we would expect an increase in the multiplier
of the incentive constraint λ (implying that the favorable bias for a high-value project is
larger and that the monetary prize is more often awarded) and a reduction in the optimal
innovation effort.
4.2 Robust mechanisms with respect to v
The optimal mechanism allocates the project on the basis of its value. In practice, this
value may be difficult to measure objectively or may be costly to verify, which in turn calls
for more robust rules. Even in this case, biasing the implementation tender still provides
an effective way of incentivizing innovators. To see this fact, in Online Appendix D, we
consider a variant of our baseline model in which: (i) the buyer, having observed the value
of the project, remains free to decide whether to implement the project or not; and (ii) the
implementation tender cannot depend on the value of the project (that is, x (v, θ) = x (θ) and
ρ (v) = ρ for all v). Obviously, the innovator has no incentive to exert any research effort
when the project is never implemented or when it is always implemented (in this latter
case, the innovator gets the same expected information rent, regardless of the value of its
proposal). However, if the project is implemented only when it is sufficiently valuable, then
it is always optimal to bias the tender in favor of the innovator.23 Interestingly, handicaps
are never optimal in this case. In addition, as long as the principal observes the value of the
project, such mechanism can be used regardless of whether this value is also observed by the
firms or can be verified by third parties such as courts.
22It can be checked that this mechanism indeed ensures that the innovator is never worse off than a pure
contractor – see Online Appendix C.23Specifically, the shadow costs are of the form K1 (θ1) < J1 (θ1) and Ki (θi) = Ji (θi) for i > 1.
23
4.3 Current practice on unsolicited proposals
Public authorities are sometimes approached directly by private companies with proposals for
developing projects even without any formal solicitation. To allow the contracting authority
to make a proper evaluation of the technical and economic feasibility of the project and
to determine whether the project is likely to be successfully implemented, the proposer
must typically submit a technical and economic feasibility study, an environmental impact
study and satisfactory information regarding the concept or technology contemplated in the
proposal. Despite the often significant cost involved in these submissions, some countries
do not allow public authorities to directly reward these unsolicited proposals. Our analysis
suggests instead that it can be optimal to reward valuable proposals through contract rights
and possibly through monetary prizes. Hodges and Dellacha (2007) describe three alternative
ways that are used in practice:
◦ Bonus system. The system gives the original project proponent a bonus in the tendering
procedure. A bonus can take many forms but most commonly involves additional points in
the score of the original proponent’s technical or financial offer. This system is, for example,
adopted in Chile and Korea. In the former, the bonus points are linked to the value of the
proposed project.
◦ Swiss challenge system. The Swiss challenge system gives the original project proponent
the right to counter-match any better offers. This system is most common in the Philippines
and is also used in Guam, India, Italy, and Taiwan. Under this procedure, the original
proposer will counter-match the lowest rival bid and win the contract whenever its cost is
less than that bid. Anticipating this, rival bidders will respond by shading their bids but
still bid above their costs. Hence, the system distorts the contract allocation in favor of the
proposer (who wins the contract for sure when its cost is less than the rivals’ costs but may
also win when its cost is above theirs).24
◦ Best and final offer system. Here, the key element is multiple rounds of tendering, in
which the original proponent is given the advantage of automatically participating in the
final round. This system is used in Argentina and South Africa.
Our analysis suggests that these mechanisms have some merit as biasing the implemen-
tation stage in favor of the innovator may indeed promote innovation. The bonus system
has the additional merit of allowing the advantage to be linked to the value of the proposed
project. Furthermore, as discussed in Section 4.2, the unconditional advantage granted to
the innovator under the Swiss challenge system and the best and final offer system can be
rationalized when the value of the project is difficult to verify. None of these systems involve
explicit handicapping.
24See Burguet and Perry (2009) for the formal analysis of the right of first refusal in a procurement
context. Their model does not involve ex ante investment, however.
24
4.4 On the Optimality of Bundling R&D and Implementation
In the practice of innovation procurement, we observe two polar approaches.
The first approach is pure bundling, wherein the firm whose project is selected also im-
plements it. This approach was, for instance, followed in US defense procurement in the
1980s, where the winner of the technical competition for the best prototype was virtually as-
sured of being awarded the follow-on defense contract (see Lichtenberg, 1990; and Rogerson,
1994). More recently, the European Procurement Directive 2014/24/EU has introduced the
so-called “innovation partnerships” for the joint procurement of R&D services and large-scale
production. As mentioned in the Introduction, in the US the National Defense Authorization
Act for 2016 recently expanded the use of Other Transaction Authority, thereby enhancing
the possibility of direct assigning the production contract to the firm that was awarded the
contract for the R&D services.
The second approach is unbundling, in which the selection of the project and its imple-
mentation are kept entirely separate; therefore, the firm to which the selected project belongs
is treated exactly in the same way as any other firm at the implementation stage. Exam-
ples of this approach include research contests, the European Pre-commercial Procurement
(PCP) model,25 and the standard approach to R&D procurement under the US federal laws
and regulations that apply to government procurement contracts.26 In these cases, firms
compete for innovative solutions at the R&D stage, and the best solution(s) may receive a
prize. The procurer does not commit itself to acquiring the resulting innovations.
Our analysis identifies specific circumstances in which the two extreme cases can be
optimal.
Corollary 3. 1. Pure bundling is optimal if, for each i, k ∈ N , ψkk = 0 and ψki =∞ if
i 6= k.
2. Unbundling is optimal if there exists N1, N2 ⊂ N with N1 ∪N2 = N and N1 ∩N2 = ∅such that for each i, k ∈ N , ψki = 0 if k ∈ N1 and i ∈ N2 and ψki = ∞ otherwise. In
this case, the optimal mechanism selects the project k from N1 with the highest value vk
and awards the implementation contract to the firm i ∈ N2 with the lowest virtual cost
Ji(θi), provided that maxk∈N1 vk ≥ mini∈N2 Ji(θi); in addition, it rewards the innovator
h ∈ N1 with the highest βh(vh), provided that it exceeds 1.
Pure bundling can be optimal when there are large economies of scope between R&D
and } {implementation, as in the case described by the condition in Corollary 3-(1). For
example, in the procurement of complex IT systems, the knowledge acquired by the software
25See EC (2007) and https://ec.europa.eu/digital-agenda/en/pre-commercial-procurement.26Part 35, Federal Acquisition Regulation (FAR), https://www.acquisition.gov/browsefar.
25
developer typically confers a considerable cost advantage for the management and upgrading
of the software. In this case, selecting the same firm for both R&D and implementation is
likely to be better. However, even in that case, our analysis suggests that the selection of
the project should be based on both value and cost considerations.
By contrast, unbundling is optimal when firms specialize in either innovation or in im-
plementation (e.g., manufacturing or construction). Corollary 3-(2) describes such a case:
firms are partitioned into two groups so that one specializes in innovation and the other
specializes in implementation.27 In this case, the optimal mechanism selects the project and
rewards the innovator from the former group according to the first-best scheme in Proposi-
tion 1 and awards the implementation contract to a firm in the second group according to
the second-best scheme in Proposition 2.
Unbundling is sometimes prescribed as an affirmative action policy toward small and
medium enterprises (SMEs). In both Europe and the US, procurement programs aimed
at stimulating R&D investment from SMEs provide for separation between the R&D stage
and the implementation stage. Funding is provided based on firms’ project proposals. The
Small Business Innovation Research (SBIR) program in the US and the Small Business
Research Initiative (SBRI) in the UK are characterized by this separation between project
selection and implementation.28 Such a policy can be justified based on Corollary 3-(2) on
the grounds that small or medium R&D firms often lack manufacturing capabilities and thus
would be at a clear disadvantage when the R&D competition is bundled with the contract
implementation. For instance, if SMEs constitute group N1 and non-SMEs constitute N2,
it is then desirable to promote research effort specifically from SMEs and ban non-SMEs
from proposing a project (as under SBIR and SBRI). Indeed, a study commissioned by the
European Commission29 finds empirical evidence that PCP (i.e., unbundling) increases both
participation by and awarding to SMEs compared to conventional joint procurement of R&D
services and supply (i.e., bundling).
A similar reasoning suggests that when base university research plays a key role in R&D
activities, separation between selection and implementation may help to promote universities’
participation. When instead innovators are also likely to play a role at the implementation
stage, unbundling is never optimal.
27While Corollary 3-(2) portrays pure implementors (firms j ∈ N2) in terms of high R&D costs, a similar
insight applies when they are productive in research (e.g., if F j is concentrated on v).28See, respectively, http://www.sbir.gov/ and https://sbri.innovateuk.org.29See Bedin, Decarolis and Iossa (2015).
26
5 Related Literature
Prizes versus property rights to motivate innovation. The issue of prizes vs. contracts
is reminiscent of the well-known debate on the effectiveness of the patent system as a source
of incentive for innovation (see Maurer and Scotchmer, 2004; and Cabral et al., 2006) for
reviews. Just as in our model, the patent system involves ex post distortion (in terms of both
too little quantity and foreclosure of competing firms), making prizes apparently preferable
(see, e.g., Kremer, 1998). However, the literature has shown that as in this paper, ex post
distortion can be an optimal way to motivate ex ante innovation. The difference lies in the
motivation for the ex post distortion. In the case of Weyl and Tirole (2012), for instance, the
supplier has private information at the ex ante (innovation) stage; the awarding of property
rights—a source of inefficiency—serves to reveal the value of innovation. In our case, private
information in the ex post implementation stage, coupled with limited liability, forces the
buyer to leave rents to the winning supplier. These rents can be harnessed as incentives for
innovation but only when the allocation of the contract rights is shifted in favor or against
the innovator as a function of the value of innovation. That is, the distortion in the allocation
of contracts rights arises as a way to incentivize innovation.
Bundling sequential tasks. Our analysis is related to the literature on whether two
tasks should be allocated to the same agent (“bundling”) or to two different agents (“un-
bundling”). The existing literature finds that this choice can be driven by problems of adverse
selection (see, e.g., Armendariz and Gollier, 1998; Ghatak, 2000), monitoring (Besley and
Coate, 1995; Armendariz, 1999; Rai and Sjostrom, 2004), moral hazard (Stiglitz, 1990; Var-
ian 1990; Holmstrom and Milgrom, 1991; Itoh, 1993), or agents’ limited liability (Laffont
and Rey, 2003). A second strand of the literature has focused specifically on sequential
tasks. Our paper is specifically related to Riordan and Sappington (1989), who highlighted
how sole sourcing (bundling) can serve as commitment device to incentivize R&D effort, by
raising the prospect of a lucrative follow-on contract. In their context, the buyer suffers from
limited commitment power and the value of the project is non-verifiable. Like them, we show
that contract rights can provide incentive for R&D effort, but we consider verifiable project
values and full commitment, thus extending the buyer’s options to the possibility that his
choice depends on the realized project value or that he commits ex ante to a given bias.
Other more recent papers have studied the role of externalities across tasks (Bennett
and Iossa, 2006), budget constraints (Schmitz, 2013), information on the ex post value of
the second task (Tamada and Tsai, 2007) or on the future cost of improving the service
provision (Hoppe and Schmitz, 2013) or competition among agents (Li and Yu, 2015). Our
paper contributes to this literature by showing that the implementation decision should
depend on the value of the proposed project(s) as well as on the supplier’s characteristics.
Full unbundling is therefore typically not optimal unless innovators and implementors form
27
distinct groups, while pure bundling is optimal only under rather specific conditions – namely,
when the innovator is in a much better position to implement its project.
Discrimination and bidding parity in auctions. Our analysis is also related to the
literature on discrimination in auctions. This subset of the literature finds it optimal to
distort the allocation to reduce the information rents accruing to the bidders: discrimination
against efficient types helps level the playing field and elicit more aggressive bids from other-
wise stronger bidders (Myerson, 1981; McAfee and McMillan, 1985). In a similar vein, when
bidders can invest in cost reduction, an ex post bias in the auction design can help foster
bidders’ ex ante investment incentives (Bag 1997) or prevent the reinforcement of asymmetry
among market participants (Arozamena and Cantillon, 2004). Likewise, manipulating the
auction rules can help motivate investment in cost reduction by an incumbent firm (Laffont
and Tirole, 1988), incentivize monitoring effort by an auditor (Iossa and Legros, 2004) or
favor the adoption of an efficient technology by an inefficient firm (Branco, 2002). We con-
tribute to this literature by showing that when investment is “cooperative” (in the sense of
Che and Hausch, 1999) and directly benefits the buyer, both favoritism and handicapping
are optimal, depending on the value of the proposed project and on the bidders’ costs.
Contests. Finally, another large literature studies the provision of incentives through
contests or tournaments. Since the seminal contributions by Tullock (1967, 1980) and
Krueger (1974) on rent-seeking and of Becker (1983) on lobbying, the framework has been
applied to many other situations, including research contests.30 This literature typically as-
sumes that agents’ efforts affect the probability of winning the contest but not the associated
reward.31 In contrast, here, the principal can reward innovators with contract rights as well
as with a monetary prize, which enables her to influence how innovators’ efforts affect their
information rents. This, in turn, allows us to analyze the optimal composition of a prize.
Dynamic Contracting. The idea of using future “rents” an agent expects to earn to
motivate her earlier effort has been developed in the dynamic contracting literature. For
example, Board (2011), Andrews and Baron (2016), Calzolari and Spagnolo (2017) and
Calzolari et al. (2015) consider moral hazard models, showing that the optimal relational
contract rewards good behavior with loyalty. However, the relational models do not allow
for contingent prizes, as we do, and cannot therefore discuss the relative value of alternative
instruments. The idea that the principal can solve short-term moral hazard problem by
committing to a long-term and potentially inefficient allocation of business is also present
in Garrett and Pavan (2012), who consider repeated adverse selection and examine the
30See Corchon (2007), Konrad (2009) and Long (2013) for surveys of the literature on contests. For
research contests, see, e.g., Che and Gale (2003) and Piccione and Tan (1996) for an analysis of R&D
investment in cost reduction followed by standard procurement auctions.31Some papers allow the reward to depend on agents’ efforts. For instance, in “winner-takes-all” races,
firms’ investments in cost reduction may affect not only the probability of winning the market, but also the
profit achieved in that case. However, this relation remains exogenously given.
28
distortions in allocation that can occur over time. They consider both the design of retention
policy (analogous to the “contract rights” here) and the use of managerial compensations
(analogous to a “prize” here) and analyze how their use evolves over time. An important
distinction lies in the main concern of the principal, which is to minimize informational rents
in their paper and to incentivize innovation in ours.
6 Conclusion
Procuring innovative projects requires incentivizing research efforts from potential suppliers
as well as implementing the selected projects efficiently. We have shown how the procurer
may optimally address these objectives by combining cash prizes and contract rights as
alternative tools for rewarding innovators.
A number of issues are worth exploring further. First, for the most part we have focused
on situations in which the value of the proposals can be contracted upon. This is a plausi-
ble assumption when, for instance, the proposal involves a prototype or when performance
measures – operational or productivity indicators, energy consumption, emissions, etc. – are
available and can be used in tender documents or when the procurer can rely on impartial
evaluation committees. In other situations (e.g., base research), however, the difficulty of
describing the project and/or non-verifiability issues may make it impossible to contract ex
ante on the ex post value of the projects. Even in the latter case, if the values of projects are
observable to the relevant parties (i.e., procurer and firms), then the non-verifiability prob-
lem can be overcome for free as the values can be elicited through an incentive scheme (see
Maskin and Tirole (1999)). The situation is different when the value of the project is private
information (e.g., only the buyer observes it). Yet, the spirit of our insights carries over
when, for instance, the procurer must use the same auction rules whenever she decides to
implement the project (see the discussion in Section 4.2). Characterizing the optimal mech-
anism under private information is beyond the scope of this paper but clearly constitutes an
interesting avenue for future research.
Second, we have ignored the costs of participating in procurement tenders. In practice,
submitting a tender bid may require tender development costs (e.g., complex estimations and
legal advice) that involve significant economic resources, in which case biasing the tender in
favor of the innovator may discourage potential suppliers from participating in the tender.
It would therefore be worth endogenizing the participation in the tender and exploring how
the optimal mechanism should be adjusted to account for these development costs. More
generally, accounting for endogenous entry is a promising research avenue.32
32For recent work on the role of discrimination in auctions with endogenous entry, see, e.g., Jehiel and
Lamy (2015).
29
Likewise, we have assumed that the procurer was benevolent. In practice, corruption
concerns and, more generally, institution design may matter, which may call for limiting the
discretion given to the procuring agency. Balancing this with the provision of innovation
incentives constitutes another promising research avenue.33
Finally, we have focused on a situation where the innovation is valuable to a single buyer
– and thus has no “market” value. An interesting extension would be to consider multiple
buyers so as to allow for the possibility that extra contractual incentives for research effort
arise from the commercialization of the innovation. Exploring the role of market forces would
also help to shed light on possible anti-competitive effects of alternative mechanisms for the
public procurement of innovation.34
33For recent work on the role of corruption in procurement auctions, see, e.g., Burguet (2015).34See the 2014 European State Aid framework for research, development and innovation (EU 2014b).
30
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35
Appendix
A Proof of Proposition 1
To solve [P − FB], we focus on the relaxed problem:
[P ′ − FB] maxx,t
Ev,θ[v∑i∈N
xi(v, θ)− ti(v, θ)|e]
subject to (LL), (MH) and
Eθ[ti(v, θ)− θixi(v, θ)] ≥ 0, ∀v, i. (IR′)
This problem is a relaxation of [P − FB] since (IR′) requires (IR) to hold only on
average. At the same time, whenever a mechanism satisfies (IR′), one can construct at least
one mechanism that satisfies (IR), without affecting other constraints. Hence, there is no
loss in restricting attention to [P ′ − FB]. To solve [P ′ − FB], we first observe that for each
i 6= 1, the constraint (IR′) must bind. If not, one can always lower the expected payment
to increase the value of the objective without tightening any constraints. Next, define
ρ1(v) := Eθ[t1(v, θ)− θ1x1(v, θ)].
Then, we can weaken [P ′ − FB] further to:
[P ′′ − FB] maxx,t
Ev,θ[∑i∈N
(v − θi)xi(v, θ)− ρ1(v)|e]
subject to
ρ1(v) ≥ 0, ∀v, (IR′′)
Eθ[∑i∈N
(v − θi)xi(v, θ)] ≥ ρ1(v), ∀v, (LL′′)
∂
∂eEv [ρ1(v)|e] ≥ c′(e). (MH ′′)
Note that the weakening occurs with the moral hazard constraint: (MH ′′) is a first-order
necessary condition of (MH).
Let ν(v), µ(v), and λ denote the multipliers for constraints (IR′′), (LL′′) and (MH ′′),
respectively. Then, the Lagrangian (more precisely its integrand) is given by:
L(v, θ, e) := [1 + µ(v)]
{∑i∈N
xi(v, θ) (v − θi)
}− ρ1(v) [1 + µ(v)− ν(v)− β (v)]− λc′ (e) ,
36
where
β (v) := λfe(v|e)f(v|e)
.
The optimal solution (eFB, xFB (v, θ) , ρFB (v) , λFB, µFB (v) , νFB (v)) must satisfy the
following necessary conditions.
First, since the Lagrangian is linear in xi’s, the optimal solution xFBi (v, θ) is as defined
in Proposition 1. Next, the Lagrangian L is also linear in ρ1(v); hence, its coefficient must
be equal to zero:
1 + µ∗(v)− β∗ (v)− ν∗(v) = 0. (1)
Next, the optimal effort eFB must satisfy
∂
∂eEv,θ[L(v, θ, e)|e]
∣∣∣∣e=eFB
= 0. (2)
Finally, complementary slackness implies that, for each v,
νFB(v)ρFB(v) = 0, (3)
µFB(v)
{Eθ[∑i∈N
xFBi (v, θ) (v − θi)− ρFB1 (v)
}= 0, (4)
and
λFB[∫
v
ρFB(v)fe(v|eFB)dv − c′(eFB)
]= 0. (5)
We first prove that λFB > 0. Suppose not. Then, βFB(v) = 0 for all v ∈ V . It then
follows from (1) that νFB(v) > 0 for all v ∈ V . By (3), this means that ρFB(v) ≡ 0. As
xi = xFBi , it then follows from (4) that for any v > θ, µFB(v) = 0. Collecting these facts
together, we conclude that
Eθ[L(v, θ, e)] = Eθ[max{0, v −mini{θi}],
which is increasing in v (and strictly so for a positive measure of v). By (MLRP ), this
means that∂
∂eEv,θ[L(v, θ, e)|e] > 0,
a contradiction to (2). We thus conclude that λFB > 0.
If v < vFB, then βFB(v) < 1, and thus 1 + µFB(v) − βFB(v) > 0. Hence, by (1),
νFB(v) > 0 and, by (3), we have ρFB(v) = 0. It in turn follows from (4) that µFB(v) = 0
provided that v > θ.
37
If instead v > vFB, then βFB(v) > 1, and thus 1− βFB(v)− νFB(v) < 0. Hence, by (1),
µFB(v) > 0. But then, by (4), we must have
ρFB(v) = Eθ
[∑i∈N
xFBi (v, θ) (v − θi)
],
as claimed in Proposition 1.
Next, we show that v < vFB < v. First, by (MLRP ), βFB (v) is strictly increasing in v,
and there exists v ∈ (v, v) such that βFB (v) = 0 (< 1); it follows that vFB > v (> v). Second,
we must have v > vFB with positive probability (i.e., λFB cannot be too small). Suppose to
the contrary that βFB(v) < 1 for all v ∈ V . Then, as argued above ρFB(v) = µFB(v) = 0
for all v ∈ V . In this case, by the convexity of c(·), we must have eFB = 0, or else we obtain
a contradiction to (5). But then, we get
L(v, θ, e) = max{0,maxθi
(v − θi)} − c′ (e) .
As the first term is increasing in v (and strictly so for a positive measure of v), and c′ (0) = 0,
we thus get a contradiction to (2).
Finally, we prove that eFB > 0. Given λFB > 0, it follows from (5) that∫v
ρFB(v)fe(v|eFB)dv = c′(eFB).
As v > vFB for a positive measure of v, the left side is strictly positive. This implies that
eFB > 0, or else the right-hand side vanishes as c′(0) = 0.
B Proof of Proposition 3
To solve [P ], we first reformulate (IC) in terms of interim allocation and payment rules. For
each i ∈ N and for any v ∈ V and any θi ∈ Θi, let Xi(v, θi) :=∫θ−i
xi (v, θ) dG−i (θ−i) and
Ti(v, θi) :=∫θ−i
ti (v, θ) dG−i (θ−i) denote the interim allocation and payment for firm i and
Ui(v, θi) := Ti(v, θi)− θiXi(v, θi) (6)
denote firm i’s expected profit. For each i ∈ N , (IC) then can be stated as
Ti(v, θi)− θiXi(v, θi) ≥ Ti(v, θ′i)− θiXi(v, θ
′i), ∀v, θi, θ′i.
The associated envelope condition then yields
Ui(v, θi) = ρi (v) +
∫ θ
θi
Xi(v, θ)dθ, (7)
38
where
ρi (v) := Ui(v, θ)
is the rent enjoyed by firm i when its cost is highest. Using (7), we can express firm i’s
expected rent as∫θi
Ui(v, θi)dGi (θi) =
∫θi
[ρi(v) +
∫ θ
θi
Xi(v, s)ds
]dGi (θi)
= ρi(v) +
∫θi
Xi(v, θi)Gi (θi)
gi (θi)dGi (θi) . (8)
For each i 6= 1, the rent ρi(v) does not help to relax any constraint and reduces the surplus
for the principal, so it is optimal to set ρi(v) = 0 for all v.
Using (6) and (8), the total expected transfer to the firms can be expressed as:∫θ
∑i∈N
ti (v, θ) dG (θ) =∑i∈N
∫θi
Ti(v, θi)dGi (θi)
=∑i∈N
∫θi
[Ui(v, θi) + θiXi(v, θi)] dGi (θi)
=∑i∈N
{ρi(v) +
∫θi
Xi(v, θi)Ji(θi)dGi (θi)
}= ρ1(v) +
∫θ
∑i∈N
xi (v, θi) Ji(θi)dG (θ) , (9)
where Ji(θi) := θi + Gi(θi)gi(θi)
denotes firm i’s virtual cost.
Substituting (9) into the principal’s objective function, we can rewrite (LL) as follows:
∀v ∈ V,∫θ
{∑i∈N
xi(v, θ) [v − Ji(θi)]
}dG(θ) ≥ ρ1(v). (LL)
Let µ(v) ≥ 0 denote the multiplier associated with this constraint.
The innovating firm’s individual rationality simplifies to
∀v ∈ V, ρ1(v) ≥ 0. (IR)
Let ν(v) ≥ 0 denote the multiplier associated with this constraint.
We next focus on the first-order condition for the effort constraint.∫v
∫θ
[ρ1(v) +
G1(θ1)
g1(θ1)x1(v, θ)
]dG(θ)fe(v|e)dv ≥ c′(e). (MH)
39
Note that we formulate the condition as a weak inequality to ensure the nonnegativity of
the multiplier. Let λ ≥ 0 be the associated multiplier.
Then, [P ] can more succinctly be reformulated as follows:
maxe,x(v,θ),ρ1(v)
∫v
{∫θ
[∑i∈N xi(v, θ) [v − Ji(θi)]
]dG(θ)− ρ1(v)
}f(v|e)dv
subject to (LL), (IR) and (MH)
The integrand of the Lagrangian is given by:
L(v, θ, e) := [1 + µ(v)]
[v − θ1 −
(1− β (v)
1 + µ(v)
)G1(θ1)
g1(θ1)
]x1(v, θ) +
∑j∈Nj 6=1
[v − Jj(θj)]xj(v, θ)
− ρ1(v) [1 + µ(v)− ν(v)− β (v)]− λc′ (e) ,
where
β (v) := λfe(v|e)f(v|e)
.
The optimal solution (e∗, x∗ (v, θ) , ρ∗1 (v) , λ∗, µ∗ (v) , ν∗ (v)) must satisfy the following nec-
essary conditions. First, observe that the Lagrangian L is linear in ρ1(v); hence, its coefficient
must be equal to zero:
1 + µ∗(v)− β∗ (v)− ν∗(v) = 0. (10)
The Lagrangian is also linear in xi’s, so the optimal allocation must satisfy, for every
i, v, θ:
x∗i (v, θ) =
{1 if i ∈ arg minj
{Kj(v, θj)
}and Ki (v, θi) ≤ v,
0 otherwise,
where
Ki(v, θi) :=
{Ji (θi)− β∗(v)
1+µ∗(v)Gi(θi)gi(θi)
if i = 1,
Ji (θi) if i 6= 1,
where β∗ (v) = λ∗ fe(v|e∗)
f(v|e∗).
Next, the optimal effort e∗ must satisfy
∂
∂e
∫v
∫θ
L(v, θ, e∗)f(v|e∗)dG (θ) dv = 0. (11)
Finally, complementary slackness implies that, for each v,
ν∗(v)ρ∗1(v) = 0, (12)
µ∗(v)
{∫θ
∑i∈N
x∗i (v, θ) [v − Ji(θi)] dG(θ)− ρ∗1(v)
}= 0, (13)
40
and
λ∗[∫
v
∫θ
[ρ∗1(v) +
G1(θ1)
g1(θ1)x∗1(v, θ)
]dG (θ) fe(v|e∗)dv − c′(e∗)
]= 0. (14)
We now provide the characterization. Consider first the case where v < θ. From (10),
Ki(v, θi) ≥ θi, and thus Ki(v, θi) and K∗i (v, θi) both yield x∗i (v, θ) = 0 for every i ∈ N ;
furthermore, (LL) and (IR) together imply
ρ∗1(v) = 0 =
∫θ
∑i∈N
x∗i (v, θ) [v − Ji(θi)] dG(θ).
Hence, the characterization of x∗i (v, θ) given in Proposition 3 is correct.
We now focus on the range v > θ. Again, there are two cases depending on the value of
v. Consider first the case v < v, where β∗ (v) < 1. Hence, 1+µ∗(v)−β∗ (v) > µ∗(v) ≥ 0, and
(10) thus implies ν∗(v) > 0. The complementary slackness condition (12) then yields ρ∗1(v) =
0. This, together with Lemma 4 (see Online Appendix B) and the complementary slackness
condition (13), implies that µ∗(v) = 0. Hence, K1(v, θ1) = J1 (θ1) − β∗ (v)G1(θ1)/g1(θ1) =
K∗1(v, θ1).
Let us now turn to the case v > v, where β∗ (v) > 1. Hence, 1− β∗ (v)− ν∗(v) < 0, and
(10) thus implies that µ∗(v) > 0; from the complementary slackness condition (13), we thus
have
ρ∗1(v) =
∫θ
∑i∈N
x∗i (v, θ) [v − Ji(θi)] dG(θ).
Suppose ν∗(v) > 0. Lemma 4 (of Online Appendix B) then implies ρ∗1(v) > 0, contradicting
the complementary slackness condition (12). Therefore, ν∗(v) = 0. It follows now from (10)
that 1 + µ∗(v) = β∗ (v). We therefore conclude that K1(v, θ1) = θ1 = K∗1(v, θ1).
The expected transfer payment T ∗i (v, θi) follows from (6) and (7), with ρ∗1(v) as described
above and ρ∗j(v) = 0 for all j 6= 1. The above characterization is valid only when the optimal
allocation is monotonic (another necessary condition from incentive compatibility). This
follows the assumption that Gi(θi)gi(θi)
is nondecreasing in θi, which implies thatK∗i (v, θi) = Ji(θi),
for i 6= 1, and
K∗1(v, θ1) = J1 (θ1)−min {1, β (v)} G1(θ1)
g1(θ1)= θ1 + max {0, 1− β (v)} G1(θ1)
g1(θ1),
are all nondecreasing in θi.
We next prove that λ∗ > 0. Suppose λ∗ = 0. Then, β∗(·) = 0, so (10) again implies that
ν∗(·) > 0 and µ∗(·) = ρ∗1(·) = 0. Hence,
L(v, θ, e∗) = max{0, v −miniJi(θi)},
41
which increases for a positive measure of v. It follows that
∂
∂e
∫ v
v
∫θ
L(v, θ, e)dG(θ)f(v|e)dv∣∣∣∣e=e∗
=
∫ v
v
∫θ
max{0, v −miniJi(θi)}dG(θ)fe(v|e∗)dv > 0,
which contradicts (11).
Next, we show that e∗ > 0. It follows from (14) and λ∗ > 0 that∫v
∫θ
[ρ1(v) +
G1(θ)
g1(θ)x∗1(v, θ)
]g(θ)dθfe(v|e)dv = c′(e).
The left-hand side is strictly positive, which implies that e∗ > 0, or else the right side vanishes
since c′(0) = 0.
42
Online AppendixNot for publication
A On the Optimality of Offering a Prize (v < v)
As mentioned, whether it is optimal to award a monetary prize (i.e., v < v) depends on how
much innovation incentives are required and on how much would already be provided by the
standard second-best auction. We show in this Online Appendix that a monetary prize is
optimal when: (i) there is either little cost heterogeneity (see Section A.1) or a large number
of firms (see Section A.2), as the procurement auction does not generate much information
rents, and thus provides little innovation incentives; or (ii) the range of project values is
large (see Section A.3), so that innovation incentives then matter a lot.
Throughout this Online Appendix, we start with an environment for which there ex-
ists an optimal mechanism with no monetary reward, and then consider variations of this
environment for which the optimal mechanism must involve a prize.
The baseline environment, for which there exists an optimal mechanism with no monetary
reward, consists of a distribution F (·|e) for the value v and a distribution Gi (·) for the cost
of each firm i ∈ N , such that ρ∗ (·) = 0, which amounts to v > v, or
λ∗ < λ :=f (v|e∗)fe (v|e∗)
, (15)
and implies that µ∗ (·) = 0. The optimal allocation is therefore such that x∗i (v, θ) = 0 for
any v ≤ θ and, for v > θ:
x∗1 (v, θ) =
{1 if K∗1 (θ1) < min {v, J2 (θ2) , ..., Jn (θn)} ,0 otherwise.
For further reference, it is useful to note that the objective of the principal, as a function of
e, can be expressed as:∫ v
v
∫ θ
θ
∑i∈N
x∗i (v, θ) [v − Ji (θi)] dG (θ) dF (v|e)
+ λ
{∫ v
v
∫ θ
θ
X∗1 (v, θ1)G1 (θ1) fe(v|e)dθ1dv − c′ (e)
},
where the innovator’s expected probability of obtaining the contract is given by:
X∗1 (v, θ1) =
∫θ−1
x∗1 (v, θ) dG−1 (θ−1) .
43
The first-order condition with respect to e yields:∫ v
v
∫ θ
θ
∑i∈N
x∗i (v, θ) [v − Ji (θi)] dG (θ) fe (v|e∗) dv
= λ
{c′′ (e∗)−
∫ v
v
∫ θ
θ
X∗1 (v, θ1)G1 (θ1) fee(v|e∗)dθ1dv
}. (16)
The optimal effort e∗ moreover satisfies the innovator’s incentive constraint c′ (e∗) = b (e∗),
where the innovator’s expected benefit is given by:
b (e) :=
∫ v
v
∫ θ
θ
X∗1 (v, θ1)G1 (θ1) fe(v|e)dθ1dv.
A.1 Reducing Cost Heterogeneity
Suppose first that costs become increasingly less heterogeneous: the cost of each firm i ∈ Nbecomes distributed according to Gm
i (θi) over the range Θmi =
[θ, θm = θ +
(θ − θ
)/m]. For
each m ∈ N∗, we will denote by em, λm, Km1 (θ1) and Xm
1 (v, θ1) the values associated with
the optimal mechanism. We now show that, for m large enough, this optimal mechanism
must include a monetary prize.
We first note that as m goes to infinity, the innovator’s effort tends to the lowest level, e:
Lemma 1. em tends to e as m goes to infinity.
Proof. The innovator’s expected benefit becomes
bm (e) :=
∫ v
v
∫ θm
θ
Xm1 (v, θ1)Gm
1 (θ1) fe(v|e)dθ1dv,
and satisfies:
|bm (e)| ≤∫ v
v
∫ θ+ θ−θm
θ
dθ1 |fe(v|e)| dv =
(θ − θ
) ∫ vv|fe(v|e)| dvm
.
Therefore, as m goes to infinity, the expected benefit bm (e) converges to 0, and the innova-
tor’s effort thus converges to the minimal effort, e. �
Furthermore:
Lemma 2. As m goes to infinity:
• The left-hand side of (16) tends to
B∞ :=
∫ v
v
(v − θ) fe (v|e) dv > 0.
44
• In the right-hand side of (16), the terms within brackets tend to c′′ (e).
Proof. The left-hand side of (16) is of the form∫ vvhm1 (v) dv, where
hm1 (v) := fe (v|em)
∫ θm
θ
∑i∈N
xmi (v, θ) [v − Ji (θi)] dGm (θ) .
Furthermore, for any v > θ, J (θ) := mini∈N {Ji (θi)} < v for m is large enough (namely, for
m such that θm < v or m >(θ − θ
)/ (v − θ)), and so
hm1 (v) = fe (v|em)
∫ θm
θ
[v − J (θ)
]dGm (θ) ,
which is bounded:
|hm1 (v)| <∣∣∣max
efe (v|e)
∣∣∣max {v − θ, 0} ,
and converges to
limm−→∞
hm1 (v) = (v − θ) fe (v|e) .
Using Lebesgue’s dominated convergence theorem, we then have:
limm−→∞
∫ v
v
hm1 (v) dv =
∫ v
v
limm−→∞
hm1 (v) dv = B∞.
We now turn to the right-hand side (16). The terms within brackets are
c′′ (em)−∫ v
v
∫ θm
θ
Xm1 (v, θ1)Gm
1 (θ1) fee(v|em)dθ1dv,
where the first term tends to c′′ (e) and the second term is of the form∫ vvhm2 (v) dv, where
hm2 (v) = fee (v|em)
∫ θm
θ
Xm1 (v, θ1)Gm
1 (θ1) dθ1
satisfies:
|hm2 (v)| < maxe|fee (v|e)|
∫ θm
θ
dθ1 =
(θ − θ
)maxe |fee (v|e)|m
and thus tends to 0 as m goes to infinity. �
To conclude the argument, suppose that the optimal mechanism never involves a prize.
Condition (16) should thus hold for any m, and in addition, the Lagrangian multiplier λm
should satisfy the boundary condition (15). We should thus have:∫ v
v
∫ θm
θ
∑i∈N
xmi (v, θ) [v − Ji (θi)] dG (θ) fe (v|em) dv
<f (v|em)
fe (v|em)
{c′′ (em)−
∫ v
v
∫ θm
θ
Xm1 (v, θ1)G1 (θ1) fee(v|em)dθ1dv
}.
45
Taking the limit as m goes to infinity, this implies:
B∞ =
∫ v
v
(v − θ) fe (v|e) dv < f (v|e)fe (v|e)
c′′ (e) ,
which is obviously violated when the return on effort is sufficiently high (e.g., c′′ (e) is low
enough).
A.2 Increasing the Number of Firms
Let us now keep the cost distributions fixed, and suppose instead that m additional firms are
introduced in the environment with the same cost distribution as the innovator: Gk (θk) =
G1 (θk) for k = n+ 1, ..., n+m. Letting again denote by em, λm, Km1 (θ1) and Xm
1 (v, θ1) the
values associated with the optimal mechanism, we now show that the optimal mechanism
must involve a prize for m large enough.
By construction, Km1 (θ1) (> θ1) > θ for any θ1 > θ, whereas the lowest Jj (θj) becomes
arbitrarily close to J1 (θ) = θ as m increases; it follows that the probability of selecting the
innovator, Xm1 (v, θ1), tends to 0 as m goes to infinity:
Lemma 3. Xm1 (v, θ1) tends to 0 as m goes to infinity.
Proof. The probability of selecting the innovator satisfies:
Xm1 (v, θ1) ≤ Pr
[Km
1 (θ1) ≤ minj=n+1,...,n+m
{J1 (θj)}]
≤ Pr
[θ1 ≤ min
j=n+1,...,n+m{J1 (θj)}
]=
[1−G1
(J−1
1 (θ1))]m
, (17)
where the second inequality stems from Km1 (θ1) ≥ θ1, and the last expression tends to 0
when m goes to infinity. �
It follows that Lemma 1 still holds, that is, the innovator’s effort tends to the lowest
level, e, as m goes to infinity. To see this, it suffices to note that the innovator’s expected
benefit, now equal to
bm (e) =
∫ v
v
∫ θ
θ
Xm1 (v, θ1)G1 (θ1) fe(v|e)dθ1dv,
satisfies:
|bm (e)| ≤∫ v
v
h (v) dv,
46
where
h (v) := |fe(v|e)|∫ θ
θ
Xm1 (v, θ1) dθ1
is bounded (by(θ − θ
)maxv,e {|fe(v|e)|}) and, from the previous Lemma, tends to 0 as m
goes to infinity. Hence, as m goes to infinity, the expected benefit bm (e) converges to 0, and
the innovator’s effort thus tends to e.
Likewise, Lemma 2 also holds; that is,
• The left-hand side of (16) tends to B∞. To see this, it suffices to follow the same steps
as before, noting that hm1 (v), now given by
hm1 (v) =
∫ θ
θ
∑i∈N
xmi (v, θ) [v − Ji (θi)] dG (θ) fe (v|em) ,
is still bounded:
|hm1 (v)| < max {v − θ, 0}∣∣∣max
efe (v|e)
∣∣∣ ,and tends to (v − θ) fe (v|e) for any v > θ:
– J (θ) = mini∈N {Ji (θi)} is almost always lower than v when m is large enough.
Indeed, for any ε > 0, we have:
Pr[J (θ) ≤ θ + ε
]≥ Pr
[min
i=n+1,...,n+m{Ji (θi)} ≤ θ + ε
]= Pr
[min
i=n+1,...,n+m{θi} ≤ J−1
1 (θ + ε)
]= 1−
[1−G1
(J−1
1 (θ + ε))]m
,
where the last expression converges to 1 as m goes to infinity. Therefore, for any
ε > 0, there exists m1 (ε) such that for any m ≥ m1 (ε),
Pr[J (θ) ≤ θ + ε
]≥ 1− ε.
– Hence, for m ≥ m1 (ε):
v − θ ≥∫ θ
θ
∑i∈N
xmi (v, θ) [v − Ji (θi)] dG (θ) ≥ (1− ε) (v − θ − ε) ,
where the right-hand side converges to v − θ as ε tends to 0.
The conclusion then follows again from Lebesgue’s dominated convergence theorem.
47
• In the right-hand side of (16), the terms within brackets tend to c′′ (e). To see this, it
suffices to note that hm2 (v), now given by
hm2 (v) = fee (v|em)
∫ θ
θ
Xm1 (v, θ1)G1 (θ1) dθ1
– is still bounded:
|hm2 (v)| < maxe|fee (v|e)|
∫ θ
θ
Xm1 (v, θ1) dθ1
≤ maxe|fee (v|e)|
∫ θ
θ
[1−G1
(J−1
1 (θ1))]m
dθ1.
– and converges to 0: Indeed, for any ε > 0,
|hm2 (v)| < maxe|fee (v|e)|
{∫ θ+ ε2
θ
dθ1 +
∫ θ
θ+ ε2
[1−G1
(J−1
1 (θ + ε))]m
dθ1
}< max
e|fee (v|e)|
{ε2
+(θ − θ
) [1−G1
(J−1
1 (θ + ε))]m}
.
But there exists m2 (ε) such that, for any m ≥ m2 (ε):(θ − θ
) [1−G1
(J−1
1 (θ1))]m ≤ ε
2,
and thus
|hm2 (v)| < maxe|fee (v|e)| ε.
– It follows that the second term converges again to 0:
limm−→∞
∫ v
θ
hm2 (v) dv =
∫ v
θ
limm−→∞
hm2 (v) dv = 0.
The conclusion follows, using the same reasoning as in Section A.1.
A.3 Increasing the Value of the Innovation
Let us now keep the supply side (number of firms and their cost distributions) fixed and
suppose instead that:
• v is initially distributed over V = [v, v]; for the sake of exposition, we assume v � θ,35
so that the innovation is always implemented.
35Namely, v > mini∈N{Ki
(v, θ)}
.
48
• For everym ∈ N∗, the value vm becomes distributed over V m = [v, vm = v +m (v − v)],
according to the c.d.f. Fm (vm|e) = F (v + (vm − v) /m|e).
As before, letting em, λm, Km1 (θ1), and Xm
1 (v, θ1) denote the values associated with the
optimal mechanism, we now show that this optimal mechanism must involve a prize for m
large enough.
We first note that the virtual costs remain invariant here: Kmi (vm, θi) = Ki (v, θi) =
Ji (θi) for i > 1 and, as
βm (vm) = λfme (vm|e)fm(vm|e)
= λfe(v|e)f(v|e)
,
we also have
Km1 (vm, θ1) = J1 (θ1)−min {βm (vm) , 1} G1 (θ1)
g1 (θ1)
= J1 (θ1)−min {β (v) , 1} G1 (θ1)
g1 (θ1)
= K1 (v, θ1) .
As by assumption, the innovation is always implemented in this variant, the probability of
obtaining the contract only depends on these virtual costs and thus also remains invari-
ant: xmi (vm, θ) = x∗i (v, θ) for any i ∈ N . It follows that, in the right-hand side of (16),
the terms within brackets also remained unchanged: using Xm1 (vm, θ1) = X∗1 (v, θ1) and
fmee (vm|e) dvm = fee (v|e) dv, we have:
c′′ (e)−∫ vm
v
∫ θ
θ
Xm1 (vm, θ1)G1 (θ1) fmee (vm|e) dθ1dv
m = Γ∗ (e) ,
where
Γ∗ (e) := c′′ (e)−∫ v
v
∫ θ
θ
X∗1 (v, θ1)G1 (θ1) fee (v|e) dθ1dv.
By contrast, the left-hand side of (16) is unbounded asm goes to infinity: using∑
i∈N x∗i (v, θ) =
1 (as by assumption, the innovation is always implemented here), fme (v|e) dvm = fe (v|e) dvand
∫ vvfe (v|e) dv = 0, we have:
∫ vm
v
∫ θ
θ
∑i∈N
xmi (vm, θ) [vm − Ji (θi)] dG (θ) fme (vm|e) dvm
=
∫ v
v
∫ θ
θ
∑i∈N
x∗i (v, θ) [v +m (v − v)− Ji (θi)] dG (θ) fe (v|e) dv
= mB∗ (e)− C∗ (e) ,
49
where:
B∗ (e) =
∫ v
v
∫ θ
θ
∑i∈N
x∗i (v, θ) vdG (θ) fe (v|e) dv =
∫ v
v
vfe (v|e) dv,
C∗ (e) =
∫ v
v
∫ θ
θ
∑i∈N
x∗i (v, θ) Ji (θi) dG (θ) fe (v|e) dv.
To conclude the argument, suppose that the optimal mechanism never involves a prize.
Condition (16) should thus hold for any m, and in addition, the Lagrangian multiplier λm
should satisfy the boundary condition (15). We should thus have:
mB∗ (e) < C∗ (e) +f (v|e)fe (v|e)
Γ∗ (e) ,
which is obviously violated for a large enough m.
B Proof of Proposition 4
As earlier, the incentive compatibility constraint can be replaced by the envelope condition:
Ui(v, θi) = ρi(v) +
∫ θ
θi
Xi(v, s)ds, ∀(v, θi) ∈ V N ×Θ,∀i ∈ N, (18)
where
Xi(v, θi) = Eθ−i
[∑k∈N
xki (v, θi, θ−i)
].
Using (18), firm i’s expected rent can be expressed as∫θi
Ui(v, θi)dGi(θi) = ρi(v) +
∫θi
Xi(v, θi)Gi(θi)
gi(θi)dGi(θi), (19)
Using this condition, we can rewrite the limited liability constraint as:
Eθ
[∑k,i∈N
xki (v, θ){vk − ψki − Ji(θi)
}]≥∑i∈N
ρi(v), ∀v ∈ V n. (LL)
Let µ(v) ≥ 0 denote the multiplier associated with this constraint.
Also, from (18), individual rationality boils down to
ρi(v) ≥ 0, ∀v ∈ V n,∀i ∈ N. (IR)
Let νi(v) ≥ 0 denote the multiplier associated with this constraint.
50
The moral hazard constraint can be replaced by the associated first-order condition,
which, using (19), can be expressed as:36
∫v
∫θ
(ρi(v) +
Gi(θi)
gi(θi)
∑k∈N
xki (v, θ)
)dG (θ) f iei
(vi|ei
)f−i
(v−i|e−i
)dv ≥ c′(ei), ∀i ∈ N.
(MH)
We formulate again these conditions as weak inequalities to ensure the nonnegativity of the
associated multipliers, which we will denote by λ = (λ1, ..., λn).
The principal’s problem can then be more succinctly reformulated as follows:
[P ] maxx,(ρi),e
Ev,θ
[∑k,i∈N x
ki (v, θ)
(vk − Ji(θi)− ψki
)−∑
i∈N ρi(v)∣∣∣ e]
subject to (LL), (IR), and (MH).
The analysis of this problem follows the same steps as for the case of a single innovator,
and we only sketch them here. The integrand of the Lagrangian is now given by:
L(v, θ, e) := [1 + µ(v)]
{∑k,i∈N
[vk − θi −
(1− βi(vi)
1 + µ(v)
)Gi(θi)
gi(θi)− ψki
]xki (v, θ)
}−∑i∈N
ρi(v)[1 + µ(v)− νi(v)− βi(vi)
]−∑i∈N
λic′(ei),
where
βi(vi) := λif ie(v
i|e)f(vi|e)
.
The first-order conditions for the monetary prize ρi(v) and for the probability xki (v, θ) yield,
respectively:
1 + µ∗(v)− ν∗i (v)− βi∗(vi) = 0, ∀v ∈ V n,∀i ∈ N, (20)
and
xk∗i (v, θ) =
{1 if vk − Ki(v, θi)− ψki ≥ max
{max(l,j)6=(k,i) v
l − Kj(v, θj)− ψlj, 0}
,
0 otherwise,(21)
where
Ki (v, θi) := Ji(θi)−βi∗(vi)
1 + µ∗(v)
Gi(θi)
gi(θi).
Note that Ki (v, θi) can be expressed as
θi +
[1− βi∗(vi)
1 + µ∗(v)
]Gi(θi)
gi(θi),
36For simplicity, we normalize the firms’ efforts in such a way that firms face the same cost c(e); any
asymmetry can, however, be accommodated through the distributions F k(vk|e).
51
where (20) and ν∗i (v) ≥ 0 together imply that the term within brackets is non-negative. It
follows that
Ki (v, θi) ≥ θi (22)
and that Ki (v, θi) increases with θi.
The complementary slackness associated with (LL) implies that for every v ∈ V n,
µ∗(v)
{Eθ
[∑k,i∈N
xk∗i (v, θ){vk − ψki − Ji(θi)
}]−∑i∈N
ρ∗i (v)
}= 0, (23)
whereas the complementary slackness associated with (IR) implies that for every i ∈ N and
every v ∈ V n,
ν∗i (v)ρ∗i (v) = 0. (24)
We now prove the following result:
Lemma 4. Fix any v such that maxk,i{vk − ψki
}> θ. We have
Eθ
[∑k,i∈N
xk∗i (v, θ)[vk − ψki − Ji(θi)
]]> 0, (25)
if either (i) n ≥ 2 or (ii) n = 1 and either v1 − ψ11 > θ or ν1 (v1) > 0.
Proof. We first focus on the case in which n ≥ 2. Fix any v such that vl − ψlj − θ > 0
for some l, j. Further, fix any k such that∑
i xki (v, θ) > 0 for a positive measure of θs (a
project that does not satisfy this property is never adopted with positive probability and
can be ignored).
Consider first the particular case in which project k is always implemented and allocated
to the same firm i: xki (v, .) = 1 (this can, for instance, happen when vk is large and ψkj is
prohibitively high for j 6= i). In that case:
Eθ
[∑i∈N
xki (v, θ)[vk − ψki − Ji(θi)
]]
=
∫ θ
θ
[vk − ψki − Ji(θi)
]dGi(θi)
=vk − ψki − θ>0,
where the inequality stems from (21), applied to θi = θ,37 and (22).
37Generically, this condition implies vk − ψki > Ki
(v, θ); we ignore here the non-generic case vk − ψki =
Ki
(v, θ).
52
Let us now turn to the case in which no firm is selected with probability 1 to implement
project k (because project k is not always implemented and/or different firms are selected
to implement it). By (21), the optimal allocation rule is then such that
Xki (v, θi) := Eθ−i
[xki (v, θi, θ−i)
]is nonincreasing in θi for all θi ≤ vk − ψki and equals zero for any θi > vk − ψki . Further, it
is strictly decreasing in θi for a positive measure of θi if Xki (v, θi) > 0, and by the choice of
k, there is at least one such firm.
Now, for every i define
Xki (v, θi) =
{zki if θi ≤ vk − ψki0 if θi > vk − ψki ,
where zki is a constant in (0, 1) chosen so that∫ θ
θ
Xki (v, θi)dGi(θi) = zkiGi(v
k − ψki ) =
∫ θ
θ
Xki (v, θi)dGi(θi).
Clearly, zki , and hence Xki (v, ·), is well defined.
We have:
Eθ
[∑i∈N
xki (v, θ)[vk − ψki − Ji(θi)
]]
=∑i
∫ θ
θ
Xki (v, θi)
[vk − ψki − Ji(θi)
]dGi(θi)
=∑i
∫ min{θ,vk−ψki }
θ
Xki (v, θi)
[vk − ψki − Ji(θi)
]dGi(θi)
>∑i
∫ min{θ,vk−ψki }
θ
Xki (v, θi)
[vk − ψki − Ji(θi)
]dGi(θi)
=∑i
zki
∫ min{θ,vk−ψki }
θ
[vk − ψki − Ji(θi)
]dGi(θi)
=∑i
zki(max{vk − ψki − θ, 0}
)≥0.
The second equality stems from the fact that Xki (v, θi) = 0 for θi > vk − ψki , and the strict
inequality follows from the fact that: (i) vk − ψki − Ji(θi) is strictly decreasing in θi; (ii) in
the relevant range[θ,min{θ, vk − ψki }
], Xk
i (v, θi) is nonincreasing in θi and, for some i, it is
53
moreover strictly decreasing in θi for a positive measure of θi; (iii) Xki (v, ·) is constant; and
(iv) ∫ min{θ,vk−ψki }
θ
Xki (v, θi)dGi(θi) =
∫ min{θ,vk−ψki }
θ
Xki (v, θi)dGi(θi).
Summing the above string of inequalities over all k, we obtain the desired result.
Next consider the case in which n = 1. In this case, X11 (v, θ1) = x1
1(v, θ1) = 1 for
K1(v1, θ1) ≤ v1 and zero otherwise. Because Xki (v, θi) is constant when it is strictly positive,
the strict inequality above does not follow from the above argument. But the strict inequality
does still hold if v1 − ψ11 > θ or if ν1 (v1) > 0.
In the former case, the last inequality above becomes strict, thus yielding the desired
result. To consider the latter case, assume without loss v1 − ψ11 ≤ θ. Because ν1 (v1) > 0,
we have β1(v1) < 1 + µ(v1), so K1 (v1, θ1) > θ1, which implies that there exists θ < v1 − ψ11
such that x11(v, θ1) = 1 for θ1 < θ and x1
1(v, θ1) = 0 for θ1 > θ. Let θ := sup{θ ≤θ|v1 − ψ1
1 − J1(θ) ≥ 0}. If θ ≤ θ, then
Eθ[x1
1(v, θ)[v1 − ψ1
1 − J1(θ1)]]
=
∫ θ
θ
[v1 − ψ1
1 − J1(θ1)]dG(θ1) > 0.
If θ > θ, the same result holds because
Eθ[x1
1(v, θ)[v1 − ψ1
1 − J1(θ1)]]
=
∫ θ
θ
[v1 − ψ1
1 − J1(θ1)]dG(θ1)
>
∫ θ
θ
[v1 − ψ1
1 − J1(θ1)]dG1(θ1) +
∫ v1−ψ11
θ
[v1 − ψ1
1 − J1(θ1)]dG1(θ1)
=
∫ v1−ψ11
θ
[v1 − ψ1
1 − J1(θ1)]dG1(θ1)
=0,
where the strict inequality holds because v1−ψ11 − J1(θ1) < 0 for θ1 ∈ (θ, v1−ψ1
1) (which in
turn holds because θ < θ < v1−ψ11), and the last equality follows from integration by parts.
�
Without loss of generality, assume n ≥ 2 (otherwise, there would be a single innovator,
a case studied earlier). There are two cases. Consider first the case in which βi(vi) < 1 for
every i ∈ N . By (20), we must then have
ν∗i (v) = 1 + µ∗(v)− βi∗(vi) > 0,
54
and the complementary slackness condition (24) thus yields ρ∗i (v) = 0 for every firm i ∈N . This, together with (25) and the complementary slackness condition (23), implies that
µ∗(v) = 0, and thus
Ki(v, θ1) = Ji(θi)− βi∗(vi)Gi(θi)
gi(θi):= K∗i (v, θ1).
Consider next the case in which maxi∈N {βi∗(vi)} > 1. Let I = arg maxi∈N {βi∗(vi)} for
the firms that have the highest βi∗(vi). Applying (20) to i ∈ I then yields
µ∗(v) = ν∗i (v) + βi∗(vi)− 1 > ν∗ı (v) ≥ 0, (26)
whereas applying (20) to firm j 6∈ I yields
1 + µ∗(v)− ν∗i (v) = βi∗(vi) > βj∗(vj) = 1 + µ∗(v)− ν∗j (v).
It follows that ν∗j (v) > ν∗i (v) ≥ 0 for i ∈ I , j 6∈ I. Therefore, by complementary slackness
(24), ρ∗j(v) = 0, so that only firms i ∈ I can receive a positive monetary prize: ρ∗j(v) = 0 for
j 6∈ I. Finally, the complementary slackness condition (23) yields
∑i∈I
ρ∗i (v) =∑i∈N
ρ∗i (v) = Eθ
[∑k,i∈N
xk∗i (v, θ){vk − ψki − Ji(θi)
}].
By Lemma 4, the total prize must be strictly positive for all v such that vk > ψki +θ for some
k, i. Given the atomlessness of Fi(·|e) for all e, I is a singleton with probability one. Hence,
for any v such that vk > ψki + θ for some k, i, and maxi{βi∗(vi)} > 1, with probability one
only one firm receives the monetary prize.
Last, we derive the characterization of the optimal allocation rule and transfers. By the
above argument, there exists at least one firm i ∈ I such that ρ∗i (v) > 0, and for that firm,
(24) yields ν∗i (v) = 0. However, then (20) applied to all j ∈ I along with the fact that
βi∗(vi) = βj∗(vj) for i, j ∈ I means that ν∗i (v) = 0 for all i ∈ I. It then follows that
1 + µ∗(v) = maxi{βi∗(vi)}.
We thus conclude that
Ki(v, θ1) = Ji(θi)−(
βi∗(vi)
maxk βk∗(vk)
)(Gi(θi)
gi(θi)
):= K∗i (v, θ1).
Finally, the expected transfers to firm i, Ti (v, θi) , can be derived from (19) using the allo-
cation described above and Ui (v, θi) = Ti (v, θi)− Eθ−i[∑k
(ψki + θi
)xk∗i (v, θ)
].
55
C Forbidding Handicaps
We explore here how the optimal mechanism is modified when handicaps are ruled out.
Specifically, we suppose that the innovator cannot be handicapped compared to the standard
second-best allocation. That is, for every v and θ:
x1(v, θ) ≥ xSB1 (v, θ), (NH)
where:
xSB1 (v, θ) :=
{1 if J1(θ1) ≤ min {v,minj 6=1 Jj(θj)} ,
0 otherwise.
Letting α(v, θ) ≥ 0 be the multiplier of the no-handicap constraint (NH), the Lagrangian
becomes
L(v, e) := [1 + µ(v)]
[v − J1 (θ1) +
β (v)
1 + µ (v)
G1(θ1)
g1(θ1)+
α (v, θ)
1 + µ (v)
]x1(v, θ) +
∑j∈Nj 6=1
[v − Jj(θj)]xj(v, θ)
− ρ1(v) [1 + µ(v)− ν(v)− β (v)]− λc′ (e) + α(v, θ)[x1(v, θ)− xSB1 (v, θ)]
and the additional complementary slackness is
α(v, θ)[x1(v, θ)− xSB1 (v, θ)
]= 0. (27)
The Lagrangian is still linear in xi’s, so the optimal allocation must satisfy, for every
i, v, θ:
xi(v, θ) =
{1 if i ∈ arg minj
{Kj(v, θj)
}and Ki (v, θi) ≤ v,
0 otherwise,
where the shadow cost is now given by:
Ki(v, θi) :=
{Ji (θi)− β(v)
1+µ(v)Gi(θi)gi(θi)
− α(v,θ)1+µ(v)
if i = 1,
Ji (θi) if i 6= 1,with β (v) := λ
fe(v|e)f(v|e)
.
When v > v, K1(v, θ1) < J1 (θ1), and we can thus ignore the constraint (NH); hence
α(v, θ) = 0, implying K1(v, θ1) = K1(v, θ1) and x1(v, θ) = x∗1(v, θ). Let us now consider the
case v < v. If α (v, θ) = 0, the above characterization yields again x1(v, θ) = x∗1(v, θ), and
v < v then implies K1(v, θ1) > J1(θ1) and thus x1(v, θ) < xSB1 (v, θ) for at least some θs,
contradicting (NH); therefore, we must have α (v, θ) > 0, and the complementary slackness
condition (27) thus implies x1(v, θ) = xSB1 (v, θ), and thus K1(v, θ1) = J1(θ1).
The other constraints are unaffected; thus the optimal effort e must satisfy
∂
∂e
∫v
∫θ
L(v, θ, e)f(v|e)dvdG (θ) = 0,
56
and complementary slackness implies that, for each v,
ν(v)ρ1(v) = 0,
µ(v)
{∫θ
∑i∈N
xi(v, θ) [v − Ji(θi)] dG(θ)− ρ1(v)
}= 0,
and
e
[∫v
∫θ
[ρ1(v) +
G1(θ)
g1(θ)x1(v, θ)
]g(θ)dθfe(v|e)dv − c′(e)
]= 0.
Going through the same steps as before and summing up, we have:
• Ruling out handicaps implies that contract rights are allocated according to the stan-
dard second-best for low-value projects: For v < v, α (v, θ) = −β (v)G1(θ1)/g1 (θ1) (> 0)
and Ki(v, θi) = Ji(θi) for all i (and thus, xi(v, θ) = xSB (θ) for all i as well).
• Ruling out handicaps has instead no impact on optimal contract rights for high-value
projects: For v > v, α (v, θ) = 0 and xi(v, θ) = x∗i (v, θ) for all i.
In addition, forbidding handicaps does not affect the size of the monetary prize when
such a prize is given:
• For v < v, ν (v) = 1− β (v) > 0 and thus ρ1 (v) = 0 and µ (v) = 0.
• For v > v, ν (v) = 0 and β (v) = 1 +µ (v), and thus K1(v, θ1) = θ1 and thus x1(v, θ) =
x∗1(v, θ), based on K1 (v, θ1) = θ1 and Ki (v, θi) = Ji (θi) for i 6= 1; it follows that
ρ1(v) =
∫θ
∑i∈N
x∗i (v, θ) [v − Ji(θi)] dG(θ) = ρ∗1 (v) .
Note however that ruling out handicaps can affect the conditions under which a prize
is given: banning handicaps alters the multiplier λ, which in turn affects the threshold v,
which is determined by the condition λfe (v|e) /f (v|e) = 1.
D Fixed allocation
We show here that our main insight carries over when the procurer is required to use the
same tender rules whenever she decides to implement the project. The optimal mechanism
relies on contract rights (possibly combined with monetary prizes) to induce the innovator
to exert effort. Indeed, as long as the project is not always implemented, it is optimal to
57
bias the implementation auction in favor of the innovator (handicaps instead should never
be used).
Specifically, we consider a setup where, should the procurer wish to implement the
project, the mechanism (x, t) ∈ ∆n × Rn cannot depend on v. We can then simply de-
note by xi(θ) the probability that firm i implements the project and by ti(θ) the transfer
payment that it receives.
The timing of the game is now as follows:
1. The buyer offers a mechanism specifying the allocation x and a payment ti to each
firm i.
2. The innovator chooses e; the value v is then realized.
3. The buyer observes v and decides whether to implement the project, in which case
firms observe their costs and decide whether to participate.
4. Participating firms report their costs, the project is allocated (or not), and transfers
are made according to the mechanism (x, t).
If the procurer decides to implement the project, firm i’s expected profit no longer de-
pends on the project value v, and can thus be written as
Ui(θi) := Ti(θi)− θiXi(θi)
where Xi(θi) :=∫θ−i
xi (θ) dG−i (θ−i) and Ti(θi) :=∫θ−i
ti (θ) dG−i (θ−i). Using incentive
compatibility, this expected profit can be expressed as
Ui(θi) = ρi +
∫ θ
θi
Xi(θ)dθ,
where
ρi := Ui(θ).
is the rent enjoyed by firm i when its cost is highest. As before, it is optimal to set ρi = 0
for i 6= 1, and thus the total expected transfer to the firms is given by∫θ
∑i∈N
ti (θ) dG (θ) = ρ1 +
∫θ
∑i∈N
xi (θi) Ji(θi)dG (θ) ,
where Ji(θi) := θi + Gi(θi)gi(θi)
denotes firm i’s virtual cost. It follows that the procurer chooses
to implement the project when:∫θ
{∑i∈N
xi(θ) [v − Ji(θi)]
}dG(θ) ≥ ρ1.
58
As the left-hand side strictly increases with v, there exists a unique v ∈ [v, v] such that this
constraint is strictly satisfied if v > v, and violated if v < v.
Obviously, if v = v, then the project is never implemented, and thus the innovator has
no incentive to provide any effort. The assumption that v > θ guarantees that this is not
optimal. Conversely, if v = v then the project is always implemented. This could be optimal
if even low-value projects were still sufficiently desirable, but implies again the innovator
has no incentive to provide any effort, as it obtains for sure the same information rents,
regardless of the realized value of the project. From now on, we will focus on the case where
the optimal threshold is interior, i.e., v ∈ (v, v). Let µ ≥ 0 denote the multiplier associated
with the above constraint for v = v:∫θ
{∑i∈N
xi(θ) [v − Ji(θi)]
}dG(θ) ≥ ρ1. (LL)
The innovating firm’s individual rationality boils down here to
ρ1 ≥ 0. (IR)
Let ν ≥ 0 denote the multiplier associated with this constraint.
Finally, the first-order condition for the effort constraint becomes:∫v≥v
∫θ
[ρ1 +
G1(θ1)
g1(θ1)x1(θ)
]dG(θ)fe(v|e)dv ≥ c′(e). (MH)
Let λ ≥ 0 be the associated multiplier.
The buyer’s problem can then be formulated as follows:
maxe,v,x(θ),ρ1
∫v≥v
{∫θ
[∑i∈N xi(θ) [v − Ji(θi)]
]dG(θ)− ρ1
}f(v|e)dv
subject to (LL), (IR) and (MH)
The Lagrangian is given by:
L =
∫v≥v
{∫θ
[∑i∈N
xi(θ) [v − Ji(θi)]
]dG(θ)− ρ1
}f(v|e)dv
+µ
[∫θ
{∑i∈N
xi(θ) [v − Ji(θi)]
}dG(θ)− ρ1
]
+νρ1 + λ
[∫v≥v
∫θ
[ρ1 +
G1(θ1)
g1(θ1)x1(θ)
]dG(θ)fe(v|e)dv − c′(e)
].
59
Re-arranging terms, it can be expressed as L =∫θL(θ, e, v)dG(θ), where
L(θ, e, v) := [1− F (v|e) + µ]
[ve − J1(θ1) +
βe
1 + µ
G1(θ1)
g1(θ1)
]x1(θ) +
∑i∈Ni 6=1
[ve − Ji(θi)]xi(θ)
− [1− F (v|e)] ρ1 (1− βe + µ− ν)− λc′(e).
where:
µ :=µ
1− F (v|e)and ν :=
ν
1− F (v|e)denote the weighted value of the Lagrangian multipliers µ and ν (weighted by the probability
of implementing the project), and:
βe := λ
∫v≥v
fe(v|e)1− F (v|e)
dv and ve :=ve + µv
1 + µ,
where
ve :=
∫v≥v
vf(v|e)
1− F (v|e)dv.
The optimal solution (e∗, v∗, x∗ (θ) , ρ∗1, λ∗, µ∗, ν∗) must satisfy the following necessary
conditions. First, observe that the Lagrangian L is linear in ρ1(v); hence, its coefficient must
be equal to zero:
1− βe∗ + µ∗ − ν∗ = 0, (28)
where µ∗ and ν∗ denote the optimal values of the weighted multipliers, and
βe∗ :=λ∗
1− F (v|e∗)
∫v≥v
fe(v|e∗)dv.
The Lagrangian is also linear in xi’s, so the optimal allocation must satisfy, for every
i, v, θ:
x∗i (θ) =
{1 if i ∈ arg minj
{Kj(θj)
}and Ki (θi) ≤ ve+µ∗v
1+µ∗,
0 otherwise,
where
Ki(θi) :=
{Ji (θi)− βe∗
1+µ∗Gi(θi)gi(θi)
if i = 1,
Ji (θi) if i 6= 1.(29)
We next prove that λ∗ > 0. Suppose λ∗ = 0, which implies βe∗ = 0. Together with (29)
and (28), this yields
L(θ, e, v∗) = [1− F (v∗|e) + µ∗]
∫θ
max{
0, ve∗ −miniJi (θi)
}dG (θ) ,
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and thus
∂
∂e
∫θ
L(θ, e, v∗)dG(θ)
∣∣∣∣e=e∗
= −Fe(v∗|e∗)∫θ
max{
0, ve∗ −miniJi (θi)
}dG(θ), (30)
where, in the right-hand side:
• The first term, −Fe(v∗|e∗), as
− Fe(v|e) > 0. (31)
for any v∗ ∈ (v, v). To see this, note that
−Fe(v|e) =∂
∂e[1− F (v|e)] =
∫v≥v
fe(v|e)dv,
where from (MLRP ), fe(v|e) > 0 for v > v and fe(v|e) < 0 for v < v. Therefore,
if v ≥ v, then∫v≥v fe(v|e)dv > 0. If instead v < v, then
∫v<v
fe(v|e)dv < 0; but by
construction, ∫v≥v
fe(v|e)dv +
∫v<v
fe(v|e)dv =∂
∂e
∫f(v|e)dv = 0,
implying again that∫v≥v fe(v|e)dv > 0.∫
θ
∑i∈N
xi(θ) [v − Ji(θi)] dG(θ) ≥ ρ1
• The second term is also positive. Indeed, we have:∫θ
max{
0, ve∗ −miniJi (θi)
}dG(θ) >
∫θ
max{
0, v −miniJi (θi)
}dG(θ)
≥∫θ
∑i∈N
x∗i (θ) [v − Ji(θi)] dG(θ)
≥ 0,
where the first inequality stems ve > v for any v∗ < v, and the last one follows from
(LL) and (IR).
It follows that the right-hand side of (30) is positive, and thus
∂
∂e
∫θ
L(θ, e, v∗)dG(θ)
∣∣∣∣e=e∗
> 0,
which violates the optimality of e∗. We thus conclude that λ∗ > 0.
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Given λ∗ > 0, (31) implies βe∗ > 0. It then follows from (29) that the innovator benefits
from a favorable bias in the allocation of the contract rights.
Finally, complementary slackness implies that, for each v,
ν∗ρ∗1 = 0, (32)
µ∗
{∫θ
∑i∈N
x∗i (θ) [v − Ji(θi)] dG(θ)− ρ∗1
}= 0, (33)
When (0 <) βe∗ < 1, we have:
1− βe∗ + µ∗ > µ∗ ≥ 0,
and (28) thus implies ν∗ > 0. The complementary slackness condition (32) then yields
ρ∗1 = 0.
When instead βe∗ > 1 we have:
1− βe∗ − ν∗ < 0,
and (28) thus implies that µ∗(v) > 0; from the complementary slackness condition (33), we
thus have
ρ∗1(v) =
∫θ
∑i∈N
x∗i (θ) [v∗ − Ji(θi)] dG(θ).
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